Rewrite $\left(\frac{7}{8}\right)^{-4}$ As A Fraction Raised To A Positive Exponent.$\left(\frac{7}{8}\right)^{-4} = \square$

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Understanding Negative Exponents

In mathematics, exponents are a way to represent repeated multiplication. A positive exponent indicates the number of times a base is multiplied by itself, while a negative exponent indicates the reciprocal of the base raised to the positive exponent. In this article, we will focus on rewriting negative exponents as fractions raised to positive exponents.

The Rule for Negative Exponents

The rule for negative exponents states that for any non-zero number a and any integer n, the following equation holds:

a^(-n) = 1 / a^n

This means that a negative exponent can be rewritten as the reciprocal of the base raised to the positive exponent.

Rewriting Negative Exponents as Fractions Raised to Positive Exponents

Now, let's apply this rule to the given expression:

$\left(\frac{7}{8}\right)^{-4}$

Using the rule for negative exponents, we can rewrite this expression as:

$\left(\frac{7}{8}\right)^{-4} = \frac{1}{\left(\frac{7}{8}\right)^4}$

Simplifying the Expression

To simplify the expression, we need to evaluate the exponent 4 on the fraction 7/8. This can be done by multiplying the numerator and denominator by themselves 4 times:

$\left(\frac{7}{8}\right)^4 = \frac{7^4}{8^4}$

$= \frac{2401}{4096}$

Now, we can substitute this value back into the original expression:

$\left(\frac{7}{8}\right)^{-4} = \frac{1}{\frac{2401}{4096}}$

Simplifying the Reciprocal

To simplify the reciprocal, we can invert the fraction by swapping the numerator and denominator:

$\frac{1}{\frac{2401}{4096}} = \frac{4096}{2401}$

Therefore, the rewritten expression is:

$\left(\frac{7}{8}\right)^{-4} = \frac{4096}{2401}$

Conclusion

In this article, we have learned how to rewrite negative exponents as fractions raised to positive exponents. We have applied this rule to the given expression $\left(\frac{7}{8}\right)^{-4}$ and simplified the resulting expression to obtain the final answer: $\frac{4096}{2401}$. This technique is useful for simplifying complex expressions and is an essential tool in mathematics.

Examples and Applications

Negative exponents are used in various mathematical applications, including:

  • Algebra: Negative exponents are used to simplify expressions and solve equations.
  • Calculus: Negative exponents are used to represent the reciprocal of a function.
  • Physics: Negative exponents are used to represent the inverse of a physical quantity, such as velocity or acceleration.

Common Mistakes to Avoid

When working with negative exponents, it's essential to remember the following:

  • Negative exponents are not the same as negative numbers. A negative exponent indicates the reciprocal of the base, while a negative number is a value less than zero.
  • Negative exponents can be rewritten as fractions raised to positive exponents. This is a fundamental property of exponents that can be used to simplify complex expressions.

Practice Problems

To practice rewriting negative exponents as fractions raised to positive exponents, try the following problems:

  1. Rewrite the expression $\left(\frac{3}{4}\right)^{-2}$ as a fraction raised to a positive exponent.
  2. Rewrite the expression $\left(\frac{2}{3}\right)^{-5}$ as a fraction raised to a positive exponent.
  3. Rewrite the expression $\left(\frac{5}{6}\right)^{-3}$ as a fraction raised to a positive exponent.

Answer Key

  1. 169\frac{16}{9}
  2. 24332\frac{243}{32}
  3. 216125\frac{216}{125}
    Frequently Asked Questions (FAQs) on Rewriting Negative Exponents ====================================================================

Q: What is a negative exponent?

A: A negative exponent is a mathematical operation that indicates the reciprocal of a base raised to a positive exponent. In other words, a negative exponent is the inverse of a positive exponent.

Q: How do I rewrite a negative exponent as a fraction raised to a positive exponent?

A: To rewrite a negative exponent as a fraction raised to a positive exponent, you can use the rule:

a^(-n) = 1 / a^n

This means that you can rewrite a negative exponent by taking the reciprocal of the base and raising it to the positive exponent.

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

A: A negative exponent is not the same as a negative number. A negative exponent indicates the reciprocal of the base, while a negative number is a value less than zero.

Q: Can I simplify a negative exponent by multiplying the numerator and denominator by the same value?

A: No, you cannot simplify a negative exponent by multiplying the numerator and denominator by the same value. This is because the negative exponent indicates the reciprocal of the base, and multiplying the numerator and denominator by the same value would change the value of the expression.

Q: How do I apply the rule for negative exponents to a fraction?

A: To apply the rule for negative exponents to a fraction, you can use the following steps:

  1. Take the reciprocal of the fraction.
  2. Raise the reciprocal to the positive exponent.

For example, if you have the expression $\left(\frac{3}{4}\right)^{-2}$, you can rewrite it as:

$\left(\frac{3}{4}\right)^{-2} = \frac{1}{\left(\frac{3}{4}\right)^2}$

$= \frac{1}{\frac{9}{16}}$

$= \frac{16}{9}$

Q: Can I use the rule for negative exponents to simplify a complex expression?

A: Yes, you can use the rule for negative exponents to simplify a complex expression. By rewriting the negative exponents as fractions raised to positive exponents, you can simplify the expression and make it easier to work with.

Q: Are there any common mistakes to avoid when working with negative exponents?

A: Yes, there are several common mistakes to avoid when working with negative exponents. These include:

  • Confusing negative exponents with negative numbers.
  • Simplifying negative exponents by multiplying the numerator and denominator by the same value.
  • Failing to take the reciprocal of the base when rewriting a negative exponent.

Q: How do I practice rewriting negative exponents as fractions raised to positive exponents?

A: To practice rewriting negative exponents as fractions raised to positive exponents, you can try the following exercises:

  • Rewrite the expression $\left(\frac{2}{3}\right)^{-5}$ as a fraction raised to a positive exponent.
  • Rewrite the expression $\left(\frac{5}{6}\right)^{-3}$ as a fraction raised to a positive exponent.
  • Rewrite the expression $\left(\frac{3}{4}\right)^{-2}$ as a fraction raised to a positive exponent.

Answer Key

  1. 24332\frac{243}{32}
  2. 216125\frac{216}{125}
  3. 169\frac{16}{9}

By following these steps and practicing with exercises, you can become more comfortable working with negative exponents and rewriting them as fractions raised to positive exponents.