Evaluate the expression. (x^-4)^3/8/(√x)^3 * (√x)^0

Answer 1

Hi Sandy.

I usually answer questions by trying to recall the tools we will find useful for the task. In this case we need five facts about powers

1. Any number to the zero power equals 1: x⁰=1
2. The product of two different powers of the same number is just the number (base) to the sum of the powers: x^ax^b=x^a+b. Example: x³x⁵=x³⁺⁵=x⁸
3. Negative powers turn numerators to denominators and denominators to numerators. Examples: x^-2=1/x² 1/x^-7=x⁷
4. Powers of powers are calculated by multiplying the powers: (x^a)^b=x^axb, example (x²)⁵=x^2x5=x¹⁰
5. And last but not least, roots are just fractional powers. So, for instance, the cubic root of a number is the number to the power 1/3.

As for the question: calculate (x^-4)^3/8/(√x)^3 * (√x)^0

By rule #1 (√x)⁰=1, so

(x^-4)^3/8/(√x)³*(√x)⁰=(x^-4)^3/8/(√x)³

Rule #4 is used to establish (x^-4)^3/8=x^-4x3/8=x^-3/2=(√x)^-3 the bit in bold due to rule #5

Therefore: (x^-4)^3/8/(√x)³=(√x)^-3/(√x)³ but rule #3 about negative powers implies (√x)^-3=1/(√x)³

So: (√x)^-3/(√x)³ = 1/(√x)³ 1/(√x)³=(1/(√x)³)²=1/(x^3/2)²=1/x^3/2x2=1/x³

Substitution back in the beginning yields

(x^-4)^3/8/(√x)³*(√x)⁰=1/x³ or x^-3 if you prefer negative powers.

I hope I have been helpful.

Answer 2

Hi Sandy,

First of all, n⁰=1, so forget about that (sqrt x)⁰ in the denominator. Now, your numerator is (x^-4)^(3/8). Recall that when you have a power raised to a power, you multiply the exponents. This gives:

(x^-4)^3/8= x^-12/8= x^-3/2

That's your numerator. Now, the expression (sqrt x)³ is equivalent to x^3/2. Remember that sqrt (x)= x^1/2, and remember the rule above. Multiply exponents when a power is raised to a power.

Now we have:

x^-3/2/x^3/2

This is the same as:

1/x^3/2/x^3/2

When you multiply by the reciprocal, this ultimately simplifies to:

1/x³

I hope this helps.