Rewrite The Expression Below As 6 To A Single Power: ( 6 2 ) 3 = \left(6^2\right)^3 = ( 6 2 ) 3 = □ \square □

Understanding Exponents and Powers

Exponents and powers are fundamental concepts in mathematics that help us simplify complex expressions and solve equations. In this article, we will focus on rewriting the expression $\left(6^2\right)^3$ as a single power of 6.

The Power of Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $6^2$ means $6 \times 6$, and $6^3$ means $6 \times 6 \times 6$. When we have an expression like $\left(6^2\right)^3$, we need to apply the rules of exponents to simplify it.

Applying the Rules of Exponents

To rewrite the expression $\left(6^2\right)^3$ as a single power of 6, we need to apply the rule of exponents that states $(a^m)^n = a^{m \times n}$. In this case, $a = 6$, $m = 2$, and $n = 3$.

Simplifying the Expression

Using the rule of exponents, we can simplify the expression as follows:

$\left(6^2\right)^3 = 6^{2 \times 3} = 6^6$

The Final Answer

Therefore, the expression $\left(6^2\right)^3$ can be rewritten as a single power of 6, which is $6^6$.

Why is this Important?

Understanding how to rewrite expressions with exponents as single powers is crucial in mathematics, particularly in algebra and calculus. It helps us simplify complex expressions, solve equations, and make calculations more efficient.

Real-World Applications

Exponents and powers have numerous real-world applications in fields such as science, engineering, and finance. For example, in physics, exponents are used to describe the behavior of particles and waves. In engineering, exponents are used to calculate stress and strain on materials. In finance, exponents are used to calculate compound interest and investment returns.

Conclusion

In conclusion, rewriting the expression $\left(6^2\right)^3$ as a single power of 6 is a simple yet important concept in mathematics. By applying the rules of exponents, we can simplify complex expressions and make calculations more efficient. Understanding exponents and powers is crucial in mathematics and has numerous real-world applications.

Additional Examples

Here are some additional examples of rewriting expressions with exponents as single powers:

• $\left(3^4\right)^2 = 3^{4 \times 2} = 3^8$
• $\left(2^3\right)^5 = 2^{3 \times 5} = 2^{15}$
• $\left(5^2\right)^4 = 5^{2 \times 4} = 5^8$

Practice Problems

Try rewriting the following expressions as single powers:

• $\left(4^3\right)^2$
• $\left(2^5\right)^3$
• $\left(3^2\right)^4$

Answer Key

• $\left(4^3\right)^2 = 4^{3 \times 2} = 4^6$
• $\left(2^5\right)^3 = 2^{5 \times 3} = 2^{15}$
• $\left(3^2\right)^4 = 3^{2 \times 4} = 3^8$
  Q&A: Exponents and Powers =============================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about exponents and powers.

Q: What is an exponent?

A: An exponent is a small number that is written above and to the right of a larger number, indicating how many times the larger number should be multiplied by itself.

Q: What is the difference between an exponent and a power?

A: An exponent is the small number that is written above and to the right of a larger number, while a power is the result of raising a number to a certain exponent.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to apply the rules of exponents. The most common rule is that $(a^m)^n = a^{m \times n}$.

Q: What is the rule for multiplying exponents?

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

Q: What is the rule for dividing exponents?

A: When dividing exponents with the same base, you subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m - n}$.

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

A: To rewrite an expression with exponents as a single power, you need to apply the rule of exponents that states $(a^m)^n = a^{m \times n}$.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base number should be multiplied by itself a certain number of times, while a negative exponent indicates that the base number should be divided by itself a certain number of times.

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

A: To simplify an expression with negative exponents, you need to rewrite the expression with positive exponents by moving the base number to the other side of the fraction.

Q: What is the rule for raising a power to a power?

A: When raising a power to a power, you multiply the exponents. For example, $(a^m)^n = a^{m \times n}$.

Q: What is the rule for raising a power to a power with a negative exponent?

A: When raising a power to a power with a negative exponent, you multiply the exponents and then take the reciprocal of the result. For example, $(a^m)^{-n} = \frac{1}{a^{m \times n}}$.

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

A: To simplify an expression with multiple exponents, you need to apply the rules of exponents in the correct order. First, simplify any expressions with the same base, and then simplify any expressions with different bases.

Q: What is the rule for simplifying an expression with multiple exponents and fractions?

A: When simplifying an expression with multiple exponents and fractions, you need to apply the rules of exponents and fractions in the correct order. First, simplify any expressions with the same base, and then simplify any expressions with different bases. Finally, simplify any fractions by multiplying the numerator and denominator by the same value.

Conclusion

In conclusion, exponents and powers are fundamental concepts in mathematics that help us simplify complex expressions and solve equations. By understanding the rules of exponents and powers, we can simplify expressions with multiple exponents and fractions, and make calculations more efficient.

Additional Resources

For more information on exponents and powers, check out the following resources:

• Khan Academy: Exponents and Powers
• Mathway: Exponents and Powers
• Wolfram Alpha: Exponents and Powers

Practice Problems

Try simplifying the following expressions with exponents:

• $\left(2^3\right)^4$
• $\left(3^2\right)^{-3}$
• $\left(4^5\right)^2 \times \left(2^3\right)^4$

Answer Key

• $\left(2^3\right)^4 = 2^{3 \times 4} = 2^{12}$
• $\left(3^2\right)^{-3} = \frac{1}{3^{2 \times -3}} = \frac{1}{3^{-6}} = 3^6$
• $\left(4^5\right)^2 \times \left(2^3\right)^4 = 4^{5 \times 2} \times 2^{3 \times 4} = 4^{10} \times 2^{12} = 2^{12} \times 4^{10} = 2^{12} \times (2^2)^{10} = 2^{12} \times 2^{20} = 2^{32}$