Solve (-7)^(-3): Calculating Negative Number with Negative Exponent

Question

(7)3=? (-7)^{-3}=\text{?}

Video Solution

Solution Steps

00:00 Solve the following problem
00:03 According to the laws of exponents, any number(A) raised to the power of(N)
00:06 equals 1 divided by the number(A) raised to the power of(-N)
00:09 Let's apply this to the question
00:11 The number(-7) becomes 1 divided by(-7)
00:14 and the exponent(-3) becomes -(-3)
00:17 A negative multiplied by a negative becomes a positive, therefore the exponent is 3
00:20 This is the solution

Step-by-Step Solution

We begin by using the power property for a negative exponent:

bn=1bn b^{-n}=\frac{1}{b^n} We apply it to the problem:

(7)3=1(7)3 (-7)^{-3}=\frac{1}{(-7)^3} We then subsequently notice that each whole number inside the parentheses is raised to a negative power (that is, the number and its negative coefficient together) When using the previously mentioned power property: We are careful to take this into account,

We then continue by simplifying the expression in the denominator of the fraction, remembering the exponentiation property for the power of terms in multiplication:

(am)n=amn (a^m)^n=a^{m\cdot n} We apply the resulting expression

1(7)3=1(17)3=1(1)373=1173=173=173 \frac{1}{(-7)^3}=\frac{1}{(-1\cdot7)^3}=\frac{1}{(-1)^3\cdot7^3}=\frac{1}{-1\cdot7^3}=\frac{1}{-7^3}=-\frac{1}{7^3}

In summary we are able to deduce that the solution to the problem is as follows:

(7)3=1(7)3=173=173 (-7)^{-3}=\frac{1}{(-7)^3}=\frac{1}{-7^3}=-\frac{1}{7^3}

Therefore, the correct answer is option B.

Answer

173 -\frac{1}{7^{3}}