16+ Square roots & Cube roots Questions for CAT with SOLUTIONS

Q1. Simplify the expression: √(√(81x^4) - 3√(9x^2))

Answer: 3x√(x - 1)

Solution: √(81x^4) = √(9^2 × x^4) = 9x^2

3√(9x^2) = 3(3x)

√(81x^4) - 3√(9x^2) = 9x^2 - 3(3x) = 9x^2 - 9x

√(9x^2 - 9x) = √(9x(x - 1)) = 3x√(x - 1)

Therefore, the simplified expression is: 3x√(x - 1)

Q2. If x^(1/3) + y^(1/3) = 5 and xy = 64, find the value of x^(2/3) + y^(2/3).

Answer: the value of x^(2/3) + y^(2/3) is approximately 6.702.

Solution: The problem gives us two equations:

x^(1/3) + y^(1/3) = 5 and xy = 64

The first step is to get rid of the fractional exponents by raising both sides of the first equation to the 3rd power:

(x^(1/3) + y^(1/3))^3 = 5^3

x + y = 125

Now we can use the second equation, xy = 64, to substitute y = 64/x into the first equation:

x + 64/x = 125

Multiplying both sides by x gives us a quadratic equation:

x^2 + 64 = 125x

x^2 - 125x + 64 = 0

To solve this, we can use the quadratic formula:

x = (125 ± √(125^2 - 4 × 1 × 64)) / (2 × 1)

x = (125 ± √15625 - 256) / 2

x = (125 ± √15369) / 2

x = (125 ± 124.05) / 2

This gives us two values for x: 249.05/2 = 124.525 and 0.95/2 = 0.475.

Since x and y are positive values, we take x = 124.525. Then, we can find y using the second equation:

y = 64/x = 64/124.525 ≈ 0.514

Finally, we can calculate x^(2/3) + y^(2/3):

x^(2/3) + y^(2/3) = (124.525)^(2/3) + (0.514)^(2/3)

≈ 5.927 + 0.775

≈ 6.702

So the value of x^(2/3) + y^(2/3) is approximately 6.702.

Q3. Evaluate: (√(7^5) + 5√7) / (3√7 - √(7^3))

Answer: the value of the expression is -32.25 - 1.25√7.

Solution: Step 1: Simplify the numerator and denominator separately.

Numerator:

√(7^5) + 5√7

= √(7^5) + 5(√7)

= √(16807) + 5(√7)

= 129 + 5(√7)

Denominator:

3√7 - √(7^3)

= 3(√7) - √(7^3)

= 3(√7) - 7(√7)

= -4(√7)

Step 2: Substitute the simplified numerator and denominator into the original expression.

(√(7^5) + 5√7) / (3√7 - √(7^3))

= (129 + 5√7) / (-4√7)

= -(129 + 5√7) / 4√7

= -32.25 - 1.25√7

Step 3: Simplify the final answer.

-32.25 - 1.25√7

Therefore, the value of (√(7^5) + 5√7) / (3√7 - √(7^3)) is -32.25 - 1.25√7.

Here's a more human-friendly explanation:

We need to evaluate the expression (√(7^5) + 5√7) / (3√7 - √(7^3)).

First, let's simplify the numerator and denominator separately.

The numerator is √(7^5) + 5√7.

√(7^5) means the 5th root of 7^5, which is the same as the square root of 7^10 or 16807. So, √(7^5) = √(16807) = 129.

The numerator then becomes 129 + 5√7.

For the denominator, 3√7 is just 3 times the square root of 7.

√(7^3) means the cube root of 7^3, which is the same as the square root of 7^6 or 7^3. So, √(7^3) = 7√7.

The denominator then becomes 3√7 - 7√7 = -4√7.

Now we can substitute these values into the original expression:

(129 + 5√7) / (-4√7)

= -(129 + 5√7) / 4√7

= -32.25 - 1.25√7

Therefore, the value of the expression is -32.25 - 1.25√7.

Q4. Solve for x: 3√(x+4) - 2√(x-2) = 6

Answer: the two values of x that satisfy the original equation are 65.5888 and 0.6512.

Solution: Step 1: Isolate the square root terms on one side of the equation.

3√(x+4) - 2√(x-2) = 6

2√(x-2) = 3√(x+4) - 6

Step 2: Square both sides of the equation to remove the square roots.

4(x-2) = (3√(x+4) - 6)^2

4(x-2) = 9(x+4) - 36√(x+4) + 36

Step 3: Expand and simplify the right-hand side.

4(x-2) = 9x + 36 - 36√(x+4) + 36

4x - 8 = 9x - 36√(x+4) + 72

-5x = -36√(x+4) + 64

Step 4: Add 36√(x+4) to both sides of the equation.

-5x + 36√(x+4) = 64

Step 5: Divide both sides by -5 to isolate x.

x - 7.2√(x+4) = -12.8

Step 6: Square both sides to remove the square root.

x^2 - 14.4x√(x+4) + 51.84(x+4) = 163.84

x^2 - 14.4x√(x+4) + 51.84x + 207.36 = 163.84

Step 7: Rearrange the terms to form a quadratic equation.

x^2 - 66.24x + 207.36 - 163.84 = 0

x^2 - 66.24x + 43.52 = 0

Step 8: Solve the quadratic equation using the quadratic formula.

x = (-(-66.24) ± √((-66.24)^2 - 4(1)(43.52))) / (2(1))

x = (66.24 ± √(4390.1776 - 174.08)) / 2

x = (66.24 ± √4216.0976) / 2

x = (66.24 ± 64.9376) / 2

x = 131.1776 / 2 or 1.3024 / 2

x = 65.5888 or 0.6512

Therefore, the solutions to the equation 3√(x+4) - 2√(x-2) = 6 are x = 65.5888 and x = 0.6512.

Here's a more human-friendly explanation:

We want to solve the equation 3√(x+4) - 2√(x-2) = 6.

First, we isolate the square root terms on one side: 2√(x-2) = 3√(x+4) - 6.

Then, we square both sides to remove the square roots: 4(x-2) = (3√(x+4) - 6)^2.

After expanding and simplifying, we get: -5x = -36√(x+4) + 64.

Adding 36√(x+4) to both sides gives: -5x + 36√(x+4) = 64.

Dividing both sides by -5 gives: x - 7.2√(x+4) = -12.8.

Squaring both sides again to remove the square root leads to a quadratic equation: x^2 - 66.24x + 43.52 = 0.

Using the quadratic formula, we find the solutions: x = 65.5888 and x = 0.6512.

So, the two values of x that satisfy the original equation are 65.5888 and 0.6512.

Q5. If (a+b)^(1/3) = 2 and (a-b)^(1/3) = 1, find the value of a^(2/3) + b^(2/3).

Answer: the value of a^(2/3) + b^(2/3) is 4.

Solution: Step 1: Raise both equations to the 3rd power to remove fractional exponents.

(a+b) = 2^3 = 8

(a-b) = 1^3 = 1

Step 2: Add the two equations to get:

2a = 9

Therefore, a = 9/2

Step 3: Substitute a = 9/2 into the first equation to find b.

(9/2 + b) = 8

b = 5/2

Step 4: Calculate a^(2/3) and b^(2/3).

a^(2/3) = (9/2)^(2/3) = 3

b^(2/3) = (5/2)^(2/3) = (25/8)^(1/3) = 5/2

Step 5: Find a^(2/3) + b^(2/3).

a^(2/3) + b^(2/3) = 3 + 5/2 = 8/2 = 4

Therefore, the value of a^(2/3) + b^(2/3) is 4.

Q6. Simplify: (27√(x^3) - 9x√x) / (3x√(x^3))

Answer: the simplified form of the expression is 6√x.

Solution: Step 1: Simplify the numerator and denominator separately.

Numerator:

27√(x^3) - 9x√x

= 27x(√x) - 9x(√x)

= 27x^(3/2) - 9x^(3/2)

= (27 - 9)x^(3/2)

= 18x^(3/2)

Denominator:

3x√(x^3)

= 3x(x)

= 3x^2

Step 2: Substitute the simplified numerator and denominator into the original expression.

(27√(x^3) - 9x√x) / (3x√(x^3))

= (18x^(3/2)) / (3x^2)

= 6x^(1/2)

Step 3: Simplify the final answer.

6x^(1/2) = 6√x

Therefore, the simplified form of (27√(x^3) - 9x√x) / (3x√(x^3)) is 6√x.

Here's a more human-friendly explanation:

We need to simplify the expression (27√(x^3) - 9x√x) / (3x√(x^3)).

First, let's simplify the numerator and denominator separately.

In the numerator, 27√(x^3) means 27 times the square root of x^3, which is the same as 27x(√x) or 27x^(3/2).

Similarly, 9x√x is the same as 9x^(3/2).

So, the numerator becomes 27x^(3/2) - 9x^(3/2) = (27 - 9)x^(3/2) = 18x^(3/2).

In the denominator, 3x√(x^3) is the same as 3x(x) or 3x^2.

Now we can substitute these values into the original expression:

(18x^(3/2)) / (3x^2) = 6x^(1/2) = 6√x

Therefore, the simplified form of the expression is 6√x.

Q7. If x^(1/3) = a and y^(1/3) = b, express (x+y)^(1/3) in terms of a and b.

Answer: (x + y)^(1/3) can be expressed as a + b when x^(1/3) = a and y^(1/3) = b.

Solution: Step 1: Raise both sides of the given equations to the power of 3.

(x^(1/3))^3 = a^3

x = a^3

(y^(1/3))^3 = b^3

y = b^3

Step 2: Substitute x = a^3 and y = b^3 into the expression (x + y)^(1/3).

(x + y)^(1/3) = (a^3 + b^3)^(1/3)

Step 3: Use the algebraic identity (a + b)^n = a^n + b^n for any real numbers a, b, and n.

(a^3 + b^3)^(1/3) = a + b

Therefore, (x + y)^(1/3) can be expressed in terms of a and b as:

(x + y)^(1/3) = a + b

Here's a more human-friendly explanation:

We are given that x^(1/3) = a and y^(1/3) = b.

First, we raise both sides of these equations to the power of 3 to get rid of the fractional exponents:

x = a^3 and y = b^3

Now, we want to express (x + y)^(1/3) in terms of a and b.

We substitute x = a^3 and y = b^3 into the expression:

(x + y)^(1/3) = (a^3 + b^3)^(1/3)

Using the algebraic identity (a + b)^n = a^n + b^n for any real numbers a, b, and n, we can simplify this:

(a^3 + b^3)^(1/3) = a + b

Therefore, (x + y)^(1/3) can be expressed as a + b when x^(1/3) = a and y^(1/3) = b.

Q8. Solve for x: √(x+6) + √(x-2) = 5

Answer: the two values of x that satisfy the original equation √(x+6) + √(x-2) = 5 are 14.96 and 2.04.

Solution: Step 1: Square both sides of the equation to remove the square roots.

(√(x+6) + √(x-2))^2 = 5^2

(√(x+6) + √(x-2))^2 = 25

Step 2: Use the identity (a + b)^2 = a^2 + 2ab + b^2 to expand the left-hand side.

(x+6) + 2√[(x+6)(x-2)] + (x-2) = 25

Step 3: Simplify the expression under the square root.

(x+6)(x-2) = x^2 - 2x + 6x - 12 = x^2 + 4x - 12

Step 4: Substitute the simplified expression into the equation.

x + 6 + 2√(x^2 + 4x - 12) + x - 2 = 25

2x + 4 + 2√(x^2 + 4x - 12) = 25

Step 5: Rearrange the terms to isolate the square root term.

2√(x^2 + 4x - 12) = 25 - 2x - 4

2√(x^2 + 4x - 12) = 21 - 2x

Step 6: Square both sides to remove the square root.

4(x^2 + 4x - 12) = (21 - 2x)^2

4x^2 + 16x - 48 = 441 - 84x + 4x^2

4x^2 - 68x + 489 = 0

Step 7: Solve the quadratic equation using the quadratic formula or factoring.

Using the quadratic formula:

x = (-(-68) ± √((-68)^2 - 4(4)(489))) / (2(4))

x = (68 ± √(4624 - 1956)) / 8

x = (68 ± √2668) / 8

x = (68 ± 51.66) / 8

x = 119.66 / 8 or 16.34 / 8

x = 14.96 or 2.04

Therefore, the solutions for the equation √(x+6) + √(x-2) = 5 are x = 14.96 and x = 2.04.

Here's a more human-friendly explanation:

We want to solve the equation √(x+6) + √(x-2) = 5.

First, we square both sides to remove the square roots: (√(x+6) + √(x-2))^2 = 5^2 = 25.

Then, we use the identity (a + b)^2 = a^2 + 2ab + b^2 to expand the left-hand side: (x+6) + 2√[(x+6)(x-2)] + (x-2) = 25.

After simplifying the expression under the square root, we get: 2x + 4 + 2√(x^2 + 4x - 12) = 25.

Rearranging the terms, we isolate the square root term: 2√(x^2 + 4x - 12) = 21 - 2x.

Squaring both sides again to remove the square root gives us a quadratic equation: 4x^2 - 68x + 489 = 0.

Using the quadratic formula or factoring, we find the solutions: x = 14.96 and x = 2.04.

Therefore, the two values of x that satisfy the original equation √(x+6) + √(x-2) = 5 are 14.96 and 2.04.

Q9. If a^(1/3) + b^(1/3) = 5 and a^(1/3) - b^(1/3) = 1, find a and b.

Answer: the values of a and b that satisfy the given conditions are a = 1386.5 and b = 810.5.

Solution: Step 1: Square both equations to remove the fractional exponents.

(a^(1/3) + b^(1/3))^2 = 5^2 = 25

a^(2/3) + 2(ab)^(1/3) + b^(2/3) = 25

(a^(1/3) - b^(1/3))^2 = 1^2 = 1

a^(2/3) - 2(ab)^(1/3) + b^(2/3) = 1

Step 2: Add the two equations to eliminate the term with (ab)^(1/3).

2a^(2/3) + 2b^(2/3) = 26

Step 3: Divide both sides by 2 to get a^(2/3) + b^(2/3) = 13.

Step 4: Cube both sides to find a + b.

(a^(2/3) + b^(2/3))^3 = 13^3

a + b = 2197

Step 5: Subtract the second equation from the first equation to find a^(2/3) - b^(2/3).

a^(2/3) - b^(2/3) = 24

Step 6: Square both sides to find a - b.

(a^(2/3) - b^(2/3))^2 = 24^2

a - b = 576

Step 7: Solve the system of two equations (a + b = 2197 and a - b = 576) to find a and b.

a + b = 2197

a - b = 576

2a = 2197 + 576

a = 1386.5

b = 2197 - a = 2197 - 1386.5 = 810.5

Therefore, a = 1386.5 and b = 810.5.

Here's a more human-friendly explanation:

We are given that a^(1/3) + b^(1/3) = 5 and a^(1/3) - b^(1/3) = 1. We want to find the values of a and b.

First, we square both equations to remove the fractional exponents: a^(2/3) + 2(ab)^(1/3) + b^(2/3) = 25 and a^(2/3) - 2(ab)^(1/3) + b^(2/3) = 1.

Adding these equations eliminates the term with (ab)^(1/3), giving us: 2a^(2/3) + 2b^(2/3) = 26. Dividing both sides by 2, we get: a^(2/3) + b^(2/3) = 13.

Cubing both sides gives us: a + b = 2197.

Next, we subtract the second equation from the first to find: a^(2/3) - b^(2/3) = 24. Squaring both sides gives us: a - b = 576.

Now we have a system of two equations: a + b = 2197 and a - b = 576.

Solving this system, we get: a = 1386.5 and b = 810.5.

Therefore, the values of a and b that satisfy the given conditions are a = 1386.5 and b = 810.5.

Q10. Evaluate: (√(81) + 3√(3))^3

Answer: the value of (√(81) + 3√(3))^3 is 3726 + 729√3.

Solution: To evaluate (√(81) + 3√(3))^3, we can follow these steps:

Step 1: Simplify the expression inside the parentheses.

√(81) = 9 (since 81 = 9^2)

3√(3) = 3 × √3 (separating the coefficient from the square root)

√(81) + 3√(3) = 9 + 3√3

Step 2: Raise the simplified expression to the power of 3 using the binomial expansion formula.

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Substituting a = 9 and b = 3√3, we get:

(9 + 3√3)^3 = 9^3 + 3(9^2)(3√3) + 3(9)(3√3)^2 + (3√3)^3

Step 3: Evaluate each term separately.

9^3 = 729

3(9^2)(3√3) = 3(81)(3√3) = 729√3

3(9)(3√3)^2 = 3(9)(9×3) = 729×3 = 2187

(3√3)^3 = 27√27 = 27×3 = 81

Step 4: Combine the terms to get the final answer.

(9 + 3√3)^3 = 729 + 729√3 + 2187 + 81

(9 + 3√3)^3 = 3726 + 729√3

Therefore, the value of (√(81) + 3√(3))^3 is 3726 + 729√3.

Here's a more human-friendly explanation:

We need to evaluate the expression (√(81) + 3√(3))^3.

First, we simplify the expression inside the parentheses:

√(81) = 9 (since 81 = 9^2)

3√(3) = 3 × √3 (separating the coefficient from the square root)

So, √(81) + 3√(3) = 9 + 3√3

Next, we raise this simplified expression to the power of 3 using the binomial expansion formula:

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Substituting a = 9 and b = 3√3, we get:

(9 + 3√3)^3 = 9^3 + 3(9^2)(3√3) + 3(9)(3√3)^2 + (3√3)^3

Evaluating each term separately:

9^3 = 729

3(9^2)(3√3) = 729√3

3(9)(3√3)^2 = 2187

(3√3)^3 = 81

Combining these terms, we get:

(9 + 3√3)^3 = 729 + 729√3 + 2187 + 81

(9 + 3√3)^3 = 3726 + 729√3

Therefore, the value of (√(81) + 3√(3))^3 is 3726 + 729√3.

Q11. Simplify: (√(x^4) - 2x^2) / (√(x^3) - x√x)

Answer: the simplified form of the expression is undefined for all values of x, except when x = 0, where it is an indeterminate form (0/0).

Solution: Step 1: Simplify the numerator and denominator separately.

Numerator:

√(x^4) - 2x^2

= x^2 - 2x^2

= -x^2

Denominator:

√(x^3) - x√x

= x^(3/2) - x^(3/2)

= 0

Step 2: Substitute the simplified numerator and denominator into the original expression.

(√(x^4) - 2x^2) / (√(x^3) - x√x)

= (-x^2) / 0

Step 3: Recognize that the expression is undefined for all values of x, except when the denominator is equal to 0.

Step 4: Set the denominator equal to 0 and solve for x.

√(x^3) - x√x = 0

x^(3/2) - x^(3/2) = 0

x^(3/2)(1 - 1) = 0

x^(3/2) = 0

x = 0

Step 5: Evaluate the numerator and denominator for x = 0.

Numerator: √(0^4) - 2(0^2) = 0 - 0 = 0

Denominator: √(0^3) - 0√0 = 0 - 0 = 0

Step 6: Apply the rule for indeterminate forms (0/0) by simplifying the expression using algebraic techniques or taking the limit.

lim(x→0) [(√(x^4) - 2x^2) / (√(x^3) - x√x)] = lim(x→0) [(-x^2) / (x^(3/2) - x^(3/2))]

= lim(x→0) [(-x^2) / (x^(3/2)(1 - 1))]

= lim(x→0) [(-x^2) / 0]

= Undefined

Therefore, the simplified form of (√(x^4) - 2x^2) / (√(x^3) - x√x) is undefined for all values of x, except when x = 0, where the expression is an indeterminate form (0/0).

Here's a more human-friendly explanation:

We need to simplify the expression (√(x^4) - 2x^2) / (√(x^3) - x√x).

First, we simplify the numerator and denominator separately:

Numerator: √(x^4) - 2x^2 = x^2 - 2x^2 = -x^2

Denominator: √(x^3) - x√x = x^(3/2) - x^(3/2) = 0

Substituting these values into the original expression, we get:

(-x^2) / 0

This expression is undefined for all values of x, except when the denominator is equal to 0.

Setting the denominator equal to 0 and solving for x, we get x = 0.

Evaluating the numerator and denominator for x = 0, we get:

Numerator: 0

Denominator: 0

This is an indeterminate form (0/0), which requires further simplification or taking the limit.

Taking the limit as x approaches 0, we get:

lim(x→0) [(-x^2) / (x^(3/2) - x^(3/2))]

= lim(x→0) [(-x^2) / (x^(3/2)(1 - 1))]

= lim(x→0) [(-x^2) / 0]

= Undefined

Therefore, the simplified form of the expression is undefined for all values of x, except when x = 0, where it is an indeterminate form (0/0).

Q12. If x^(2/3) = a and y^(2/3) = b, express (xy)^(1/3) in terms of a and b.

Answer: when x^(2/3) = a and y^(2/3) = b, (xy)^(1/3) can be expressed as (ab)^2.

Solution: Step 1: Raise both sides of the given equations to the power of 3/2 to remove the fractional exponents.

We are given that x^(2/3) = a and y^(2/3) = b. We want to express (xy)^(1/3) in terms of a and b.

First, we raise both sides of the given equations to the power of 3/2 to remove the fractional exponents:

x = a^3 and y = b^3

Now, we substitute these values into the expression (xy)^(1/3):

(xy)^(1/3) = (a^3 × b^3)^(1/3)

= (a^3b^3)^(1/3)

= (ab)^2

Therefore, when x^(2/3) = a and y^(2/3) = b, (xy)^(1/3) can be expressed as (ab)^2.

In other words, if we know the values of a and b, where a = x^(2/3) and b = y^(2/3), then we can find the value of (xy)^(1/3) by simply taking the product of a and b, and then squaring it.

Q13. Solve for x: 4√(x+3) - 3√(x-1) = 8

Answer: the two values of x that satisfy the original equation are 368.0341 and 1.9959.

Solution: Step 1: Isolate the square root terms on one side of the equation.

We want to solve the equation 4√(x+3) - 3√(x-1) = 8.

First, we isolate the square root terms on one side: 3√(x-1) = 4√(x+3) - 8.

Then, we square both sides to remove the square roots: 9(x-1) = (4√(x+3) - 8)^2.

After expanding and simplifying, we get: 9x - 9 = 16(x+3) - 128√(x+3) + 64.

Continuing, we have: 9x - 9 = 16x + 48 - 128√(x+3) + 64.

This simplifies to: -7x = -128√(x+3) + 113.

Adding 128√(x+3) to both sides gives: -7x + 128√(x+3) = 113.

Dividing both sides by -7 gives: x - 18.29√(x+3) = -16.14.

Squaring both sides again to remove the square root leads to: x^2 - 36.58x√(x+3) + 334.45(x+3) = 260.45.

Simplifying further, we get: x^2 - 36.58x√(x+3) + 334.45x + 1001.35 = 260.45.

Rearranging the terms, we form a quadratic equation: x^2 - 370.03x + 740.9 = 0.

Using the quadratic formula, we find the solutions: x = (-(-370.03) ± √(((-370.03)^2 - 4(1)(740.9))) / (2(1)).

This simplifies to: x = (370.03 ± √(136921.6009 - 2963.6)) / 2.

Further simplifying, we get: x = (370.03 ± √133958.0009) / 2.

This leads to: x = (370.03 ± 366.0381) / 2.

Therefore, x = 736.0681 / 2 or 3.9919 / 2.

Finally, we find: x = 368.0341 or 1.9959.

Therefore, the solutions to the equation 4√(x+3) - 3√(x-1) = 8 are x = 368.0341 and x = 1.9959.

Here's a more human-friendly explanation:

We want to solve the equation 4√(x+3) - 3√(x-1) = 8.

First, we isolate the square root terms on one side: 3√(x-1) = 4√(x+3) - 8.

Then, we square both sides to remove the square roots: 9(x-1) = (4√(x+3) - 8)^2.

After expanding and simplifying, we get: -7x = -128√(x+3) + 113.

Adding 128√(x+3) to both sides gives: -7x + 128√(x+3) = 113.

Dividing both sides by -7 gives: x - 18.29√(x+3) = -16.14.

Squaring both sides again to remove the square root leads to a quadratic equation: x^2 - 370.03x + 740.9 = 0.

Using the quadratic formula, we find the solutions: x = 368.0341 and x = 1.9959.

So, the two values of x that satisfy the original equation are 368.0341 and 1.9959.

Q14. If (a^(1/3))^3 = 8 and (b^(1/3))^3 = 27, find the value of (ab)^(1/3).

Answer: the value of (ab)^(1/3) is 6.

Solution: Step 1: Simplify the given expressions to find the values of a and b.

We are given that (a^(1/3))^3 = 8 and (b^(1/3))^3 = 27. We want to find the value of (ab)^(1/3).

First, we simplify the given expressions to find the values of a and b:

(a^(1/3))^3 = 8 means a = 8

(b^(1/3))^3 = 27 means b = 27

Now, we substitute these values into the expression (ab)^(1/3):

(ab)^(1/3) = (8 × 27)^(1/3)

= (216)^(1/3)

= 6

Therefore, the value of (ab)^(1/3) is 6.

Q15. Simplify: (3√x - 2√y) / (√(xy)) , given x = 9 and y = 16

Answer: the simplified form of the expression is 1/12.

Solution: Step 1: Substitute the given values of x and y into the expression.

We need to simplify the expression (3√x - 2√y) / (√(xy)) when x = 9 and y = 16.

First, we substitute the given values of x and y into the expression:

x = 9, y = 16

(3√x - 2√y) / (√(xy))

= (3√9 - 2√16) / (√(9 × 16))

= (3(3) - 2(4)) / (√144)

= (9 - 8) / 12

= 1 / 12

Therefore, the simplified form of the expression is 1/12.

Q16. If x^(1/2) = a and y^(1/3) = b, express (x^(2/3) * y)^(1/6) in terms of a and b.

Answer: when x^(1/2) = a and y^(1/3) = b, (x^(2/3) * y)^(1/6) can be expressed as a^(2/3) * b.

Solution: To express (x^(2/3) * y)^(1/6) in terms of a and b, where x^(1/2) = a and y^(1/3) = b, we can follow these steps:

Given information:

x^(1/2) = a

y^(1/3) = b

Step 1: Raise both sides of the given equations to the appropriate powers to remove the fractional exponents.

(x^(1/2))^2 = a^2

x = a^4

(y^(1/3))^3 = b^3

y = b^3

Step 2: Substitute x = a^4 and y = b^3 into the expression (x^(2/3) * y)^(1/6).

(x^(2/3) * y)^(1/6) = ((a^4)^(2/3) * b^3)^(1/6)

= (a^(8/3) * b^3)^(1/6)

= (a^(8/3) * b^3)^(1/6)

= (a^(4/3) * b^2)^(1/2)

= (a^(4/3) * b^2)^(1/2)

Step 3: Use the given information to express the final result in terms of a and b.

(a^(4/3) * b^2)^(1/2) = a^(2/3) * b

Therefore, when x^(1/2) = a and y^(1/3) = b, (x^(2/3) * y)^(1/6) can be expressed as a^(2/3) * b.

Here's a more human-friendly explanation:

We are given that x^(1/2) = a and y^(1/3) = b. We want to express (x^(2/3) * y)^(1/6) in terms of a and b.

First, we raise both sides of the given equations to the appropriate powers to remove the fractional exponents:

x = a^4 and y = b^3

Next, we substitute these values into the expression (x^(2/3) * y)^(1/6):

(x^(2/3) * y)^(1/6) = ((a^4)^(2/3) * b^3)^(1/6)

= (a^(8/3) * b^3)^(1/6)

= (a^(8/3) * b^3)^(1/6)

= (a^(4/3) * b^2)^(1/2)

= (a^(4/3) * b^2)^(1/2)

Using the given information, we can express the final result in terms of a and b:

(a^(4/3) * b^2)^(1/2) = a^(2/3) * b

Therefore, when x^(1/2) = a and y^(1/3) = b, (x^(2/3) * y)^(1/6) can be expressed as a^(2/3) * b.

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