Simplify The Following Expressions:A. $2^{-4} =$B. $3 A^{-1} =$C. $c \times D^{-2} =$

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In mathematics, simplifying expressions is an essential skill that helps us to solve problems efficiently and accurately. It involves rewriting complex expressions in a simpler form, making it easier to understand and work with. In this article, we will simplify three given expressions: $2^{-4}$, $3a^{-1}$, and $c \times d^{-2}$.

Simplifying Negative Exponents

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Negative exponents can be simplified by using the rule $a^{-n} = \frac{1}{a^n}$. This rule states that a negative exponent can be rewritten as a fraction with the base raised to the positive exponent in the denominator.

Example A: Simplifying $2^{-4}$

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Using the rule $a^{-n} = \frac{1}{a^n}$, we can simplify $2^{-4}$ as follows:

$2^{-4} = \frac{1}{2^4}$

To evaluate $2^4$, we need to multiply 2 by itself 4 times:

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

Now, we can rewrite $2^{-4}$ as:

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

Example B: Simplifying $3a^{-1}$

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Using the rule $a^{-n} = \frac{1}{a^n}$, we can simplify $3a^{-1}$ as follows:

$3a^{-1} = \frac{3}{a}$

Example C: Simplifying $c \times d^{-2}$

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Using the rule $a^{-n} = \frac{1}{a^n}$, we can simplify $c \times d^{-2}$ as follows:

$c \times d^{-2} = c \times \frac{1}{d^2}$

To simplify this expression further, we can rewrite it as:

$c \times d^{-2} = \frac{c}{d^2}$

Simplifying Fractional Exponents

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Fractional exponents can be simplified by using the rule $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. This rule states that a fractional exponent can be rewritten as a radical with the base raised to the numerator and the denominator as the index.

Example A: Simplifying $2^{\frac{-4}{1}}$

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Using the rule $a^{\frac{m}{n}} = \sqrt[n]{a^m}$, we can simplify $2^{\frac{-4}{1}}$ as follows:

$2^{\frac{-4}{1}} = \sqrt[1]{2^{-4}}$

Since the index is 1, we can simplify this expression further as:

$2^{\frac{-4}{1}} = 2^{-4}$

Using the rule $a^{-n} = \frac{1}{a^n}$, we can simplify $2^{-4}$ as:

$2^{-4} = \frac{1}{2^4}$

To evaluate $2^4$, we need to multiply 2 by itself 4 times:

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

Now, we can rewrite $2^{-4}$ as:

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

Example B: Simplifying $3a^{\frac{-1}{1}}$

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Using the rule $a^{\frac{m}{n}} = \sqrt[n]{a^m}$, we can simplify $3a^{\frac{-1}{1}}$ as follows:

$3a^{\frac{-1}{1}} = 3 \times \sqrt[1]{a^{-1}}$

Since the index is 1, we can simplify this expression further as:

$3a^{\frac{-1}{1}} = 3a^{-1}$

Using the rule $a^{-n} = \frac{1}{a^n}$, we can simplify $3a^{-1}$ as:

$3a^{-1} = \frac{3}{a}$

Example C: Simplifying $c \times d^{\frac{-2}{1}}$

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Using the rule $a^{\frac{m}{n}} = \sqrt[n]{a^m}$, we can simplify $c \times d^{\frac{-2}{1}}$ as follows:

$c \times d^{\frac{-2}{1}} = c \times \sqrt[1]{d^{-2}}$

Since the index is 1, we can simplify this expression further as:

$c \times d^{\frac{-2}{1}} = c \times d^{-2}$

Using the rule $a^{-n} = \frac{1}{a^n}$, we can simplify $c \times d^{-2}$ as:

$c \times d^{-2} = c \times \frac{1}{d^2}$

To simplify this expression further, we can rewrite it as:

$c \times d^{-2} = \frac{c}{d^2}$

Conclusion

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Simplifying expressions is an essential skill in mathematics that helps us to solve problems efficiently and accurately. By using the rules for negative and fractional exponents, we can simplify complex expressions and make them easier to understand and work with. In this article, we simplified three given expressions: $2^{-4}$, $3a^{-1}$, and $c \times d^{-2}$. We used the rules $a^{-n} = \frac{1}{a^n}$ and $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ to simplify these expressions and rewrite them in a simpler form.

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In the previous article, we simplified three given expressions: $2^{-4}$, $3a^{-1}$, and $c \times d^{-2}$. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q: What is the rule for simplifying negative exponents?

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A: The rule for simplifying negative exponents is $a^{-n} = \frac{1}{a^n}$. This rule states that a negative exponent can be rewritten as a fraction with the base raised to the positive exponent in the denominator.

Q: How do I simplify a negative exponent with a coefficient?

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A: To simplify a negative exponent with a coefficient, you can use the rule $a^{-n} = \frac{1}{a^n}$. For example, to simplify $3a^{-1}$, you can rewrite it as $\frac{3}{a}$.

Q: What is the rule for simplifying fractional exponents?

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A: The rule for simplifying fractional exponents is $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. This rule states that a fractional exponent can be rewritten as a radical with the base raised to the numerator and the denominator as the index.

Q: How do I simplify a fractional exponent with a coefficient?

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A: To simplify a fractional exponent with a coefficient, you can use the rule $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, to simplify $3a^{\frac{-1}{1}}$, you can rewrite it as $3 \times \sqrt[1]{a^{-1}}$.

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

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A: A negative exponent is a number raised to a negative power, while a fractional exponent is a number raised to a power that is a fraction. For example, $2^{-4}$ is a negative exponent, while $2^{\frac{-4}{1}}$ is a fractional exponent.

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

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A: To simplify an expression with multiple exponents, you can use the rules for negative and fractional exponents. For example, to simplify $c \times d^{-2}$, you can rewrite it as $\frac{c}{d^2}$.

Q: What are some common mistakes to avoid when simplifying expressions?

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A: Some common mistakes to avoid when simplifying expressions include:

• Not using the correct rule for negative or fractional exponents
• Not simplifying the expression fully
• Not checking for any errors in the calculation

Q: How can I practice simplifying expressions?

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A: You can practice simplifying expressions by working through examples and exercises in a math textbook or online resource. You can also try simplifying expressions on your own and checking your work with a calculator or online tool.

Q: What are some real-world applications of simplifying expressions?

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A: Simplifying expressions has many real-world applications, including:

• Calculating interest rates and investments
• Determining the area and perimeter of shapes
• Solving problems in physics and engineering
• Working with algebraic equations and inequalities

By understanding the rules for simplifying negative and fractional exponents, you can simplify complex expressions and make them easier to understand and work with.