Evaluate an expression with negative exponents

Question

((¹/₂)^-3) -1 : need a step-by-step process on how we obtain the answer, 7.

Answer

STEP 1:    To simplify the given expression:we should first simplify the term,and then subtract 1 from the obtained answer. We note that this term has negative exponent "–3".

The negative exponent rule states that any nonzero number raised to a negative power equals its reciprocal raised to the positive power. For instance, x^-3 can be rewritten as:
Now, let us rewrite the term,
using the negative exponent rule.

We write the reciprocal of ¹/₂ first, and then raise it to the positive power 3.

STEP 2:   Now, we are left with the expression 23 –1. Let us recall the exponential notation to simplify this expression.

For any natural number n means:
So, here it means that 2 is to be multiplied 3 times. We then subtract 1 from 2³.