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Chapter 5: Problem 53

\(8^{-2}\)

Short Answer

Expert verified
\(8^{-2} = \frac{1}{64}\)

Step by step solution

01

Understand the Negative Exponent

A negative exponent means that the base is on the wrong side of the fraction line. To move the base to the correct side, we need to invert the base and change the exponent to positive.
02

Invert the Base

Since the base is 8 and the exponent is -2, we invert the base to get \[8^{-2} = \frac{1}{8^2}\]
03

Calculate the Positive Exponent

Calculate the positive exponent of 8, which is \[8^2 = 8 \times 8 = 64\]
04

Write the Final Answer

Substitute the value from Step 3 back into the fraction to get the final answer: \[8^{-2} = \frac{1}{64}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent.
It is written as \(a^b\), where \(a\) is the base and \(b\) is the exponent.
The exponent tells us how many times to multiply the base by itself.
For instance, in the expression \(8^2\), 8 is the base and 2 is the exponent.
We calculate it as: \[ 8 \times 8 = 64 \]
Exponentiation is fundamental in algebra and appears in various mathematical problems.
It helps simplify expressions and solve equations efficiently.
Fraction Inversion
Fraction inversion is a critical concept when dealing with negative exponents.
A negative exponent means that the base is on the incorrect side of the fraction line.
For example, for \(8^{-2}\), the term can be inverted to move the base to the other side: \[ 8^{-2} = \frac{1}{8^2} \]
By inverting the base, we convert the negative exponent into a positive exponent.
This way, calculations become simpler since we are used to working with positive exponents.
Positive Exponents
Positive exponents are more intuitive since they directly represent repeated multiplication.
For instance, \(8^2\) means multiplying 8 by itself: \[ 8^2 = 8 \times 8 = 64 \]
Once a negative exponent is converted to its positive counterpart through fraction inversion,
you perform the regular exponentiation calculation.
In the example of \(8^{-2}\), after inverting the base, it becomes \(8^2\).
We then calculate the result as above, resulting in 64.
Finally, we place it under 1, giving us the answer: \[ 8^{-2} = \frac{1}{64} \]
Working with positive exponents makes solving such problems straightforward.

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Most popular questions from this chapter

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A square is a rectangle in which the lengths of all four sides are equal. If \(s=\) length of a side, write a polynomial expression in \(s\) that represents the perimeter.

Problem: Simplify: \((5 x-2)(x-3)\) Incorrect Answer: \((5 x-2)(x-3)\) $$ \begin{aligned} &=5 x(x)+5 x(-3)-2 x-2(3) \\ &=5 x^{2}-15 x-2 x-6 \\ &=5 x^{2}-17 x-6 \end{aligned} $$
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