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Simplify the following expression.

Simplify the following expression.-example-1
User Chuck Claunch
by
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1 Answer

22 votes
22 votes

Answer:

1/64

Explanation:

Simplify the following:

1/(4^(11/3)×4^(-2/3))

Hint: | Simplify radicals.

1/4^(2/3) = 1/(4^2)^(1/3) = 1/(2^4)^(1/3) = (1/2^(1/3))/2:

1/(4^(11/3)×(1/2^(1/3))/2)

Hint: | Multiply numerator and denominator of 1/(2 2^(1/3)) by 2^(2/3).

Rationalize the denominator. 1/(2 2^(1/3)) = 1/(2 2^(1/3))×(2^(2/3))/(2^(2/3)) = (2^(2/3))/(2×2):

1/(4^(11/3)×(2^(2/3))/(2×2))

Hint: | Multiply 2 and 2 together.

2×2 = 4:

1/(4^(11/3)×(2^(2/3))/4)

Hint: | Separate the exponent of 4^(-11/3) into integer and fractional parts.

4^(-11/3) = 4^(1/3 - 12/3) = 4^(-12/3)×4^(1/3):

(4^(-12/3) 4^(1/3))/((2^(2/3))/4)

Hint: | Divide 12 by 3.

12/3 = (3 (-4))/3 = 4:

(4^(-4) 4^(1/3))/((2^(2/3))/4)

Hint: | Compute 4^4 by repeated squaring.

4^4 = (4^2)^2:

(4^(1/3))/((2^(2/3))/4) 1/(4^2)^2

Hint: | Evaluate 4^2.

4^2 = 16:

(4^(1/3))/(16^2×(2^(2/3))/4)

Hint: | Evaluate 16^2.

| 1 | 6

× | 1 | 6

| 9 | 6

1 | 6 | 0

2 | 5 | 6:

(4^(1/3))/(256×(2^(2/3))/4)

Hint: | Simplify radicals.

4^(1/3) = (2^2)^(1/3):

1/(256×(2^(2/3))/4) 2^(2/3)

Hint: | Write (2^(2/3))/(256×(2^(2/3))/4) as a single fraction.

Multiply the numerator by the reciprocal of the denominator, (2^(2/3))/(256×(2^(2/3))/4) = (2^(2/3))/256×4/2^(2/3):

(2^(2/3)×4)/(256×2^(2/3))

Hint: | Cancel common terms in the numerator and denominator of (2^(2/3)×4)/(256×2^(2/3)).

(2^(2/3)×4)/(256×2^(2/3)) = (2^(2/3))/(2^(2/3))×4/256 = 4/256:

4/256

Hint: | Reduce 4/256 to lowest terms. Start by finding the GCD of 4 and 256.

The gcd of 4 and 256 is 4, so 4/256 = (4×1)/(4×64) = 4/4×1/64 = 1/64:

Answer: 1/64

User Roman Ivanov
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2.6k points