Evaluate The Following Expression For $g = -2$.$g^{-3}$g^{-3} =$ $\square$ (Type An Integer Or A Simplified Fraction.)

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Understanding the Problem

When evaluating the expression g−3g^{-3}g−3g^{-3}, we need to substitute the value of gg as −2-2 and simplify the expression. This involves understanding the properties of exponents and how to handle negative exponents.

Properties of Exponents

To evaluate the expression, we need to recall the properties of exponents. Specifically, we need to remember that when we multiply two numbers with the same base, we add their exponents. In this case, we have g−3g^{-3}g−3g^{-3}, which can be simplified using the property ama^mana^n = am+na^{m+n}.

Simplifying the Expression

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

g^{-3}$g^{-3} = $g^{-3+(-3)}$ = $g^{-6}$ ## Substituting the Value of $g$ Now that we have simplified the expression, we can substitute the value of $g$ as $-2$: $g^{-6}$ = $(-2)^{-6}$ ## Evaluating the Expression To evaluate the expression $(-2)^{-6}$, we need to remember that a negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. In this case, we can rewrite $(-2)^{-6}$ as $\frac{1}{(-2)^6}$. ## Simplifying the Expression Using the property of exponents, we can simplify the expression as follows: $\frac{1}{(-2)^6}$ = $\frac{1}{(-2)^6}$ = $\frac{1}{64}$ ## Conclusion In conclusion, when evaluating the expression $g^{-3}$g^{-3}$ for $g = -2$, we simplified the expression using the properties of exponents and substituted the value of $g$ to get the final result of $\frac{1}{64}$. ## Key Takeaways * When evaluating expressions with exponents, we need to recall the properties of exponents and how to handle negative exponents. * We can simplify expressions with exponents by adding or subtracting the exponents when multiplying or dividing numbers with the same base. * A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. ## Final Answer The final answer is $\boxed{\frac{1}{64}}$.<br/> # Q&A: Evaluating Expressions with Exponents ## Frequently Asked Questions ### Q: What is the rule for multiplying numbers with the same base? A: When multiplying numbers with the same base, we add their exponents. For example, $a^m$a^n$ = $a^{m+n}$. ### Q: How do I handle negative exponents? A: A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, $a^{-m}$ = $\frac{1}{a^m}$. ### Q: What is the rule for dividing numbers with the same base? A: When dividing numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n}$ = $a^{m-n}$. ### Q: Can I simplify expressions with exponents by combining like terms? A: Yes, you can simplify expressions with exponents by combining like terms. For example, $a^m$a^n$ + $a^m$a^p$ = $a^m(a^n + a^p)$. ### Q: How do I evaluate expressions with negative bases and exponents? A: To evaluate expressions with negative bases and exponents, you need to remember that a negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, $(-a)^m$ = $(-1)^m$a^m$. ### Q: Can I use the order of operations to evaluate expressions with exponents? A: Yes, you can use the order of operations to evaluate expressions with exponents. For example, $2^3 + 3^2$ = $8 + 9$ = $17$. ### Q: How do I simplify expressions with fractional exponents? A: To simplify expressions with fractional exponents, you need to remember that $a^{\frac{m}{n}}$ = $(a^m)^{\frac{1}{n}}$ = $\sqrt[n]{a^m}$. ### Q: Can I use the properties of exponents to simplify expressions with radicals? A: Yes, you can use the properties of exponents to simplify expressions with radicals. For example, $\sqrt[n]{a^m}$ = $a^{\frac{m}{n}}$. ### Q: How do I evaluate expressions with exponents and fractions? A: To evaluate expressions with exponents and fractions, you need to remember that you can multiply or divide fractions by multiplying or dividing their numerators and denominators. For example, $\frac{a^m}{a^n}$ = $a^{m-n}$. ### Q: Can I use the properties of exponents to simplify expressions with absolute values? A: Yes, you can use the properties of exponents to simplify expressions with absolute values. For example, $|a^m|$ = $a^m$ if $a$ is positive, and $|a^m|$ = $-a^m$ if $a$ is negative. ### Q: How do I evaluate expressions with exponents and absolute values? A: To evaluate expressions with exponents and absolute values, you need to remember that you can simplify expressions with absolute values by removing the absolute value sign and evaluating the expression inside. For example, $|a^m|$ = $a^m$ if $a$ is positive, and $|a^m|$ = $-a^m$ if $a$ is negative. ## Conclusion In conclusion, evaluating expressions with exponents requires a good understanding of the properties of exponents and how to handle negative exponents, fractional exponents, and absolute values. By following the rules and properties of exponents, you can simplify expressions and evaluate them correctly. ## Key Takeaways * When evaluating expressions with exponents, you need to recall the properties of exponents and how to handle negative exponents, fractional exponents, and absolute values. * You can simplify expressions with exponents by adding or subtracting the exponents when multiplying or dividing numbers with the same base. * A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. * You can use the order of operations to evaluate expressions with exponents. * You can use the properties of exponents to simplify expressions with radicals and absolute values. ## Final Answer The final answer is $\boxed{\frac{1}{64}}$.