rule of multiplication of powers

Oct 13, 2025 by magicalceylontours.com 0 views

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Introduction

In mathematics, indices are a fundamental concept used to represent the power or exponent of a number. When dealing with indices, it's essential to simplify expressions and express them with positive indices. This article will guide you through the process of simplifying and expressing the given expression with positive indices.

The Given Expression

The given expression is:

$\frac{5 a^{-1} b^3 c^{-4}}{2 x^5 y^{-8}}$

Simplifying the Expression

To simplify the expression, we need to apply the rules of indices. The first step is to separate the numerator and denominator.

Separating the Numerator and Denominator

$\frac{5 a^{-1} b^3 c^{-4}}{2 x^5 y^{-8}} = \frac{5}{2} \cdot \frac{a^{-1}}{1} \cdot \frac{b^3}{1} \cdot \frac{c^{-4}}{1} \cdot \frac{1}{x^5} \cdot \frac{1}{y^{-8}}$

Applying the Rule of Negative Indices

When a number is raised to a negative power, it can be rewritten as the reciprocal of the number raised to the positive power. In this case, we can rewrite $a^{-1}$ as $\frac{1}{a}$ , $c^{-4}$ as $\frac{1}{c^4}$ , and $y^{-8}$ as $\frac{1}{y^8}$ .

$\frac{5}{2} \cdot \frac{a^{-1}}{1} \cdot \frac{b^3}{1} \cdot \frac{c^{-4}}{1} \cdot \frac{1}{x^5} \cdot \frac{1}{y^{-8}} = \frac{5}{2} \cdot \frac{1}{a} \cdot b^3 \cdot \frac{1}{c^4} \cdot \frac{1}{x^5} \cdot \frac{1}{y^8}$

Applying the Rule of Multiplication of Powers

When multiplying powers with the same base, we add the exponents. In this case, we can add the exponents of $b$ , $c$ , $x$ , and $y$ .

$\frac{5}{2} \cdot \frac{1}{a} \cdot b^3 \cdot \frac{1}{c^4} \cdot \frac{1}{x^5} \cdot \frac{1}{y^8} = \frac{5}{2} \cdot \frac{b^3}{a} \cdot \frac{1}{c^4 x^5 y^8}$

Simplifying the Expression Further

We can simplify the expression further by combining the constants in the numerator.

$\frac{5}{2} \cdot \frac{b^3}{a} \cdot \frac{1}{c^4 x^5 y^8} = \frac{5b^3}{2ac^4 x^5 y^8}$

Expressing with Positive Indices

To express the simplified expression with positive indices, we need to rewrite the negative indices as positive indices.

Rewriting Negative Indices as Positive Indices

We can rewrite $a^{-1}$ as $a^1$ , $c^{-4}$ as $c^4$ , and $y^{-8}$ as $y^8$ .

$\frac{5b^3}{2ac^4 x^5 y^8} = \frac{5b^3}{2a \cdot c^4 \cdot x^5 \cdot y^8}$

Expressing with Positive Indices

We can express the expression with positive indices by rewriting the negative indices as positive indices.

$\frac{5b^3}{2a \cdot c^4 \cdot x^5 \cdot y^8} = \frac{5b^3}{2a \cdot c^4 \cdot x^5 \cdot y^8}$

Conclusion

In this article, we simplified and expressed the given expression with positive indices. We applied the rules of indices, separated the numerator and denominator, and applied the rule of negative indices. We also applied the rule of multiplication of powers and simplified the expression further. Finally, we expressed the simplified expression with positive indices by rewriting the negative indices as positive indices.

Key Takeaways

To simplify an expression with indices, we need to apply the rules of indices.
We can separate the numerator and denominator to simplify the expression.
We can rewrite negative indices as positive indices by taking the reciprocal of the number raised to the positive power.
We can apply the rule of multiplication of powers to add the exponents of powers with the same base.
We can simplify the expression further by combining the constants in the numerator.
We can express the simplified expression with positive indices by rewriting the negative indices as positive indices.

Frequently Asked Questions

Q: What are indices in mathematics?

A: Indices are a fundamental concept in mathematics used to represent the power or exponent of a number.

Q: How do we simplify an expression with indices?

A: We can simplify an expression with indices by applying the rules of indices, separating the numerator and denominator, and applying the rule of negative indices.

Q: How do we express an expression with positive indices?

A: We can express an expression with positive indices by rewriting the negative indices as positive indices.

Q: What is the rule of negative indices?

A: The rule of negative indices states that a number raised to a negative power can be rewritten as the reciprocal of the number raised to the positive power.

Q: What is the rule of multiplication of powers?

A: The rule of multiplication of powers states that when multiplying powers with the same base, we add the exponents.

References

[1] Khan Academy. (n.d.). Indices. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7d7d/x2f5f7d7d/x2f5f7d7d
[2] Math Open Reference. (n.d.). Indices. Retrieved from https://www.mathopenref.com/indices.html
[3] Purplemath. (n.d.). Exponents and Powers. Retrieved from https://www.purplemath.com/modules/exponent.htm

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Introduction

In our previous article, we discussed how to simplify and express an expression with positive indices. In this article, we will answer some frequently asked questions related to simplifying and expressing with positive indices.

Q&A

Q: What are indices in mathematics?

A: Indices are a fundamental concept in mathematics used to represent the power or exponent of a number. They are used to describe the magnitude or size of a number.

Q: How do we simplify an expression with indices?

A: To simplify an expression with indices, you need to apply the rules of indices, separate the numerator and denominator, and apply the rule of negative indices. This will help you to rewrite the expression in a simpler form.

Q: How do we express an expression with positive indices?

A: To express an expression with positive indices, you need to rewrite the negative indices as positive indices. This can be done by taking the reciprocal of the number raised to the positive power.

Q: What is the rule of negative indices?

A: The rule of negative indices states that a number raised to a negative power can be rewritten as the reciprocal of the number raised to the positive power. For example, $a^{-1}$ can be rewritten as $\frac{1}{a}$ .

Q: What is the rule of multiplication of powers?

A: The rule of multiplication of powers states that when multiplying powers with the same base, you add the exponents. For example, $a^2 \cdot a^3 = a^{2+3} = a^5$ .

Q: How do we handle exponents with the same base?

A: When you have exponents with the same base, you can add the exponents by applying the rule of multiplication of powers. For example, $a^2 \cdot a^3 = a^{2+3} = a^5$ .

Q: Can we simplify an expression with multiple negative indices?

A: Yes, you can simplify an expression with multiple negative indices by applying the rule of negative indices and the rule of multiplication of powers. For example, $\frac{a^{-1} \cdot b^{-2}}{c^{-3}} = \frac{\frac{1}{a} \cdot \frac{1}{b^2}}{\frac{1}{c^3}} = \frac{1}{a} \cdot \frac{1}{b^2} \cdot c^3 = \frac{c^3}{ab^2}$ .

Q: How do we handle fractions with negative indices?

A: When you have fractions with negative indices, you can rewrite the negative indices as positive indices by taking the reciprocal of the number raised to the positive power. For example, $\frac{a^{-1}}{b^{-2}} = \frac{\frac{1}{a}}{\frac{1}{b^2}} = \frac{1}{a} \cdot \frac{b^2}{1} = \frac{b^2}{a}$ .

Q: Can we simplify an expression with multiple fractions?

A: Yes, you can simplify an expression with multiple fractions by applying the rule of negative indices and the rule of multiplication of powers. For example, $\frac{a^{-1} \cdot b^{-2}}{c^{-3} \cdot d^{-4}} = \frac{\frac{1}{a} \cdot \frac{1}{b^2}}{\frac{1}{c^3} \cdot \frac{1}{d^4}} = \frac{1}{a} \cdot \frac{1}{b^2} \cdot c^3 \cdot d^4 = \frac{c^3 \cdot d^4}{ab^2}$ .

Conclusion

In this article, we answered some frequently asked questions related to simplifying and expressing with positive indices. We hope that this article has provided you with a better understanding of how to simplify and express expressions with positive indices.

Key Takeaways

Indices are a fundamental concept in mathematics used to represent the power or exponent of a number.
To simplify an expression with indices, you need to apply the rules of indices, separate the numerator and denominator, and apply the rule of negative indices.
To express an expression with positive indices, you need to rewrite the negative indices as positive indices.
The rule of negative indices states that a number raised to a negative power can be rewritten as the reciprocal of the number raised to the positive power.
The rule of multiplication of powers states that when multiplying powers with the same base, you add the exponents.

References

[1] Khan Academy. (n.d.). Indices. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7d7d/x2f5f7d7d/x2f5f7d7d
[2] Math Open Reference. (n.d.). Indices. Retrieved from https://www.mathopenref.com/indices.html
[3] Purplemath. (n.d.). Exponents and Powers. Retrieved from https://www.purplemath.com/modules/exponent.htm