Not rewriting the expression using the rule for negative exponents

Oct 13, 2025 by wbrum.org 45 views

Understanding Exponents and Negative Exponents

In mathematics, exponents are a shorthand way of representing repeated multiplication. For example, $6^3$ means $6 \times 6 \times 6$ . When we have a negative exponent, it means we are dealing with the reciprocal of the base raised to the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$ .

The Given Expression $$6^{-3}$$

The given expression $$6^{-3}$$ can be rewritten using the rule for negative exponents. This means that $$6^{-3} = \frac{1}{6^3}$$.

Evaluating the Options

Now, let's evaluate the options given:

A. $-6^3$

This option is incorrect because the negative sign is outside the exponentiation, which means it is not equivalent to $$6^{-3}$$.

B. $3^6$

This option is also incorrect because it is not equivalent to $$6^{-3}$$. The base is different, and the exponent is also different.

C. $\sqrt[3]{6}$

This option is incorrect because it is not equivalent to $$6^{-3}$$. The cube root of 6 is not the same as $$6^{-3}$$.

D. $\left(\frac{1}{6}\right)^3$

This option is correct because it is equivalent to $$6^{-3}$$. When we raise a fraction to a power, we raise the numerator and the denominator to that power. Therefore, $$\left(\frac{1}{6}\right)^3 = \frac{1^3}{63} = \frac{1}{6^3} = 6^{-3}$$.

Conclusion

In conclusion, the correct answer is D. $\left(\frac{1}{6}\right)^3$ . This is equivalent to $$6^{-3}$$ because it follows the rule for negative exponents and the rule for raising fractions to a power.

Understanding the Concept of Negative Exponents

Negative exponents are an important concept in mathematics, and understanding them can help us simplify expressions and solve problems more easily. By recognizing that $$a^{-n} = \frac{1}{a^n}$$, we can rewrite expressions with negative exponents in a more manageable form.

Real-World Applications of Negative Exponents

Negative exponents have many real-world applications, including:

Chemistry: In chemistry, negative exponents are used to represent the concentration of a solution. For example, a solution with a concentration of $$10^{-3} M$$ means that there are $$10^{-3}$$ moles of solute per liter of solution.
Physics: In physics, negative exponents are used to represent the decay of a quantity over time. For example, the half-life of a radioactive substance is the time it takes for the substance to decay to half its original amount, which can be represented using negative exponents.
Engineering: In engineering, negative exponents are used to represent the efficiency of a system. For example, the efficiency of a machine can be represented using negative exponents, which can help engineers design more efficient systems.

Conclusion

In conclusion, negative exponents are an important concept in mathematics, and understanding them can help us simplify expressions and solve problems more easily. By recognizing that $$a^{-n} = \frac{1}{a^n}$$, we can rewrite expressions with negative exponents in a more manageable form. Negative exponents have many real-world applications, including chemistry, physics, and engineering.

Frequently Asked Questions

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

A: A positive exponent represents repeated multiplication, while a negative exponent represents the reciprocal of the base raised to the positive exponent.

Q: How do I evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, you can rewrite it using the rule for negative exponents: $$a^{-n} = \frac{1}{a^n}$$.

Q: What are some real-world applications of negative exponents?

A: Negative exponents have many real-world applications, including chemistry, physics, and engineering.

Final Thoughts

Understanding Negative Exponents

Negative exponents are a fundamental concept in mathematics, and understanding them can help us simplify expressions and solve problems more easily. In this article, we will answer some frequently asked questions about negative exponents.

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

A: A positive exponent represents repeated multiplication, while a negative exponent represents the reciprocal of the base raised to the positive exponent.

Example:

$6^3$ means $6 \times 6 \times 6$
$6^{-3}$ means $\frac{1}{6^3}$

Q: How do I evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, you can rewrite it using the rule for negative exponents: $$a^{-n} = \frac{1}{a^n}$$.

Example:

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

Q: What are some real-world applications of negative exponents?

A: Negative exponents have many real-world applications, including:

Chemistry: In chemistry, negative exponents are used to represent the concentration of a solution. For example, a solution with a concentration of $$10^{-3} M$$ means that there are $$10^{-3}$$ moles of solute per liter of solution.
Physics: In physics, negative exponents are used to represent the decay of a quantity over time. For example, the half-life of a radioactive substance is the time it takes for the substance to decay to half its original amount, which can be represented using negative exponents.
Engineering: In engineering, negative exponents are used to represent the efficiency of a system. For example, the efficiency of a machine can be represented using negative exponents, which can help engineers design more efficient systems.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent by rewriting it using the rule for negative exponents: $$a^{-n} = \frac{1}{a^n}$$.

Example:

$\frac{1}{2^{-3}} = 2^3 = 8$

Q: How do I add or subtract expressions with negative exponents?

A: When adding or subtracting expressions with negative exponents, you can combine the exponents by adding or subtracting the exponents.

Example:

$\frac{1}{2^{-3}} + \frac{1}{2^{-2}} = 2^3 + 2^2 = 8 + 4 = 12$

Q: Can I multiply or divide expressions with negative exponents?

A: Yes, you can multiply or divide expressions with negative exponents by multiplying or dividing the exponents.

Example:

$\frac{1}{2^{-3}} \times \frac{1}{2^{-2}} = 2^3 \times 2^2 = 8 \times 4 = 32$

Q: What are some common mistakes to avoid when working with negative exponents?

A: Some common mistakes to avoid when working with negative exponents include:

Not rewriting the expression using the rule for negative exponents: Make sure to rewrite the expression using the rule for negative exponents: $$a^{-n} = \frac{1}{a^n}$$.
Not combining the exponents: When adding or subtracting expressions with negative exponents, make sure to combine the exponents by adding or subtracting the exponents.
Not multiplying or dividing the exponents: When multiplying or dividing expressions with negative exponents, make sure to multiply or divide the exponents.

Conclusion

Final Thoughts

Negative exponents are a fundamental concept in mathematics, and understanding them can help us simplify expressions and solve problems more easily. By recognizing that $$a^{-n} = \frac{1}{a^n}$$, we can rewrite expressions with negative exponents in a more manageable form. Negative exponents have many real-world applications, including chemistry, physics, and engineering. By following the rules for negative exponents and avoiding common mistakes, we can work with negative exponents with confidence.