Problem 45 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ 8^{-2} $$

Step-by-Step Solution

Answer

Question: Convert the given expression with negative exponent into an expression with only positive exponents: $$8^{-2}$$ Answer: $$\frac{1}{64}$$

Step 1: Identify the base and the exponent

In the given expression, the base is 8 and the exponent is -2: $$8^{-2}$$

Step 2: Apply the negative exponent rule

The negative exponent rule states that any number with a negative exponent can be represented as the reciprocal of the same number with a positive exponent. So we can rewrite the expression as: $$\frac{1}{8^2}$$

Step 3: Calculate the final result

Now we just need to calculate the result of the expression with the positive exponent. $$\frac{1}{8^2} = \frac{1}{64}$$ The expression with only positive exponents: $$\frac{1}{64}$$.

Key Concepts

Understanding Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (like addition and multiplication). They do not have an "equals" sign, which differentiates them from equations.
For example, the expression \(8^{-2}\) is a mathematical expression consisting of a base (8) and an exponent (-2). This expression means that we are working with an exponential function, where the base number is raised to the power of the exponent.
This kind of expression helps us simplify complex calculations and represent large or small values efficiently.

Grasping the Concept of Reciprocal

The reciprocal of a number is essentially one divided by that number. So, if we have a number \( x \), its reciprocal is \( \frac{1}{x} \). Understanding this is crucial when dealing with negative exponents.

• Negative Exponent: When a base has a negative exponent, like \(8^{-2}\), it means we need to find the reciprocal of the base with the positive value of the exponent.
• Example: \( 8^{-2} \) becomes \( \frac{1}{8^2} \), making the base positive and putting it in the denominator.

This reciprocal approach helps transform expressions into simpler forms without altering their values.

Working with Positive Exponents

In math, positive exponents indicate how many times you need to multiply a number by itself. Using positive exponents makes expressions simpler and more practical to work with.
For the expression \(8^{-2}\), converting it using the negative exponent rule leads to \(\frac{1}{8^2}\). Here, \(8^2\) means multiplying 8 by itself, resulting in \(64\). Therefore, the expression \(\frac{1}{8^2}\) becomes \(\frac{1}{64}\).
Using positive exponents allows us to work with whole numbers, which simplifies the process of solving and understanding mathematical problems.

Comprehending Exponent Rules

Exponent rules are essential for simplifying expressions and ensuring calculations are correctly carried through. They provide foundational guidelines for operating with powers of numbers:

• Negative Exponent Rule: A negative exponent \(a^{-n}\) can be rewritten as \(\frac{1}{a^n}\), turning the negative exponent part positive by using the reciprocal.
• Multiplying Like Bases: \(a^m \cdot a^n = a^{m+n}\). When you multiply terms with the same base, you add the exponents.
• Power of a Power: \((a^m)^n = a^{m \cdot n}\), meaning that when raising an exponent to another exponent, you multiply the powers.

Understanding these and other exponent rules not only aids in solving expression representations like \(8^{-2}\) but also builds a strong foundation in algebra.