6. Rewrite Each Expression As A Single Power, Then Evaluate. A) $5^2 \times 5^3$ B) $(-6)^3 \times (-6)^3$ C) $8^1 \times 8^2$7. Write Each Expression As A Product Of Two Powers, And Then As A Single Power. A)

Introduction

Exponential expressions 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 each expression as a single power and then evaluating it. We will also explore the process of writing each expression as a product of two powers and then as a single power.

Rewriting Expressions as a Single Power

a) $5^2 \times 5^3$

When we multiply two powers with the same base, we can add their exponents. In this case, the base is 5, and the exponents are 2 and 3.

Step 1: Rewrite the expression as a single power

$5^2 \times 5^3 = 5^{2+3} = 5^5$

Step 2: Evaluate the expression

$5^5 = 3125$

b) $(-6)^3 \times (-6)^3$

When we multiply two powers with the same base, we can add their exponents. In this case, the base is -6, and the exponents are 3 and 3.

Step 1: Rewrite the expression as a single power

$(-6)^3 \times (-6)^3 = (-6)^{3+3} = (-6)^6$

Step 2: Evaluate the expression

$(-6)^6 = 46656$

c) $8^1 \times 8^2$

When we multiply two powers with the same base, we can add their exponents. In this case, the base is 8, and the exponents are 1 and 2.

Step 1: Rewrite the expression as a single power

$8^1 \times 8^2 = 8^{1+2} = 8^3$

Step 2: Evaluate the expression

$8^3 = 512$

Writing Expressions as a Product of Two Powers

a) $2^4$

To write this expression as a product of two powers, we can use the fact that $a^m = a^{m/n} \times a^{(m-n)/n}$.

Step 1: Rewrite the expression as a product of two powers

$2^4 = 2^{4/1} = 2^4 \times 2^{(4-4)/1} = 2^4 \times 2^0$

Step 2: Simplify the expression

$2^4 \times 2^0 = 2^4 \times 1 = 2^4 = 16$

b) $3^5$

To write this expression as a product of two powers, we can use the fact that $a^m = a^{m/n} \times a^{(m-n)/n}$.

Step 1: Rewrite the expression as a product of two powers

$3^5 = 3^{5/1} = 3^5 \times 3^{(5-5)/1} = 3^5 \times 3^0$

Step 2: Simplify the expression

$3^5 \times 3^0 = 3^5 \times 1 = 3^5 = 243$

c) $4^3$

To write this expression as a product of two powers, we can use the fact that $a^m = a^{m/n} \times a^{(m-n)/n}$.

Step 1: Rewrite the expression as a product of two powers

$4^3 = 4^{3/1} = 4^3 \times 4^{(3-3)/1} = 4^3 \times 4^0$

Step 2: Simplify the expression

$4^3 \times 4^0 = 4^3 \times 1 = 4^3 = 64$

Conclusion

In this article, we have explored the process of rewriting exponential expressions as a single power and then evaluating them. We have also learned how to write each expression as a product of two powers and then as a single power. By understanding these concepts, we can simplify complex mathematical expressions and solve various problems with ease.

Key Takeaways

• When multiplying two powers with the same base, we can add their exponents.
• To write an expression as a product of two powers, we can use the fact that $a^m = a^{m/n} \times a^{(m-n)/n}$.
• Simplifying exponential expressions can help us solve complex mathematical problems with ease.

Final Thoughts

Q: What is the rule for multiplying two powers with the same base?

A: When multiplying two powers with the same base, we can add their exponents. For example, $a^m \times a^n = a^{m+n}$.

Q: How do I rewrite an expression as a single power?

A: To rewrite an expression as a single power, we can use the rule for multiplying two powers with the same base. For example, $a^m \times a^n = a^{m+n}$.

Q: What is the difference between $a^m$ and $a^n$?

A: $a^m$ and $a^n$ are two different powers with the same base $a$. The exponent $m$ and $n$ are different, which means the powers are different.

Q: Can I add the exponents of two powers with different bases?

A: No, you cannot add the exponents of two powers with different bases. For example, $a^m \times b^n = a^m \times b^n$, not $a^{m+n} \times b^n$.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, we can use the rule $a^{-m} = \frac{1}{a^m}$. For example, $a^{-m} = \frac{1}{a^m}$.

Q: Can I simplify an expression with fractional exponents?

A: Yes, you can simplify an expression with fractional exponents. For example, $a^{m/n} = (a^m)^{1/n}$.

Q: How do I rewrite an expression as a product of two powers?

A: To rewrite an expression as a product of two powers, we can use the rule $a^m = a^{m/n} \times a^{(m-n)/n}$. For example, $a^m = a^{m/n} \times a^{(m-n)/n}$.

Q: What is the difference between $a^m$ and $a^{m/n}$?

A: $a^m$ and $a^{m/n}$ are two different powers with the same base $a$. The exponent $m$ and $m/n$ are different, which means the powers are different.

Q: Can I simplify an expression with multiple bases?

A: Yes, you can simplify an expression with multiple bases. For example, $a^m \times b^n = (a^m) \times (b^n)$.

Q: How do I evaluate an expression with multiple bases?

A: To evaluate an expression with multiple bases, we can use the rule $a^m \times b^n = (a^m) \times (b^n)$. For example, $a^m \times b^n = (a^m) \times (b^n)$.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying exponential expressions. We have covered topics such as multiplying two powers with the same base, rewriting expressions as a single power, and simplifying expressions with negative and fractional exponents. By understanding these concepts, we can simplify complex mathematical expressions and solve various problems with ease.

Key Takeaways

• When multiplying two powers with the same base, we can add their exponents.
• To rewrite an expression as a single power, we can use the rule for multiplying two powers with the same base.
• Simplifying exponential expressions can help us solve complex mathematical problems with ease.

Final Thoughts

Simplifying exponential expressions is a crucial skill in mathematics, and understanding the rules and concepts outlined in this article can help us become proficient in solving complex mathematical problems. By practicing and applying these concepts, we can master the art of simplifying exponential expressions and become confident in our ability to solve a wide range of mathematical problems.