Rewriting Using a Single Positive Exponent

Oct 13, 2025 by miodeya.com 52 views

Introduction

Exponents are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on rewriting expressions using a single positive exponent. We will use the example of $8^{-8} \cdot 8^5$ to demonstrate the steps involved in simplifying exponents.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $8^3$ can be written as $8 \cdot 8 \cdot 8$ . The exponent $3$ indicates that the base $8$ is multiplied by itself $3$ times. When we have a negative exponent, it means that we are taking the reciprocal of the base raised to the positive exponent. For instance, $8^{-3}$ is equivalent to $\frac{1}{8^3}$ .

Rewriting Using a Single Positive Exponent

Now, let's apply this understanding to the given expression $8^{-8} \cdot 8^5$ . Our goal is to rewrite this expression using a single positive exponent.

Step 1: Apply the Product of Powers Property

The product of powers property states that when we multiply two powers with the same base, we can add their exponents. In this case, we have:

8^{-8} \cdot 8^5 = 8^{-8+5}

Step 2: Simplify the Exponent

Now, we can simplify the exponent by combining the two terms:

8^{-8+5} = 8^{-3}

Step 3: Rewrite Using a Positive Exponent

Since we want to rewrite the expression using a single positive exponent, we can take the reciprocal of the base raised to the positive exponent:

8^{-3} = \frac{1}{8^3}

Step 4: Evaluate the Expression

Finally, we can evaluate the expression by calculating the value of $8^3$ :

\frac{1}{8^3} = \frac{1}{512}

Conclusion

In this article, we have demonstrated how to rewrite an expression using a single positive exponent. We started with the expression $8^{-8} \cdot 8^5$ and applied the product of powers property to simplify the exponent. We then took the reciprocal of the base raised to the positive exponent to rewrite the expression using a single positive exponent. Finally, we evaluated the expression to obtain the final answer.

Tips and Tricks

When simplifying exponents, always apply the product of powers property first.
When rewriting an expression using a single positive exponent, take the reciprocal of the base raised to the positive exponent.
Make sure to evaluate the expression by calculating the value of the base raised to the exponent.

Common Mistakes

Failing to apply the product of powers property when simplifying exponents.
Not taking the reciprocal of the base raised to the positive exponent when rewriting an expression using a single positive exponent.
Not evaluating the expression by calculating the value of the base raised to the exponent.

Real-World Applications

Understanding how to simplify exponents is crucial in various real-world applications, such as:

Calculating interest rates and investments
Determining the area and volume of shapes
Modeling population growth and decay
Solving problems in physics and engineering

Practice Problems

Simplify the expression $2^{-4} \cdot 2^3$ using a single positive exponent.
Rewrite the expression $3^{-2} \cdot 3^4$ using a single positive exponent.
Evaluate the expression $\frac{1}{4^2}$ .

Answer Key

$2^{-4+3} = 2^{-1} = \frac{1}{2}$
$3^{-2+4} = 3^2 = 9$
$\frac{1}{4^2} = \frac{1}{16}$