Examples of Simplifying Expressions Using Exponents

Oct 13, 2025 by windailys.com 120 views

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Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication of a number. In the expression $\left(12 k^2 m^3 n^6\right)^2$ , we have a base of $12 k^2 m^3 n^6$ and an exponent of $2$ . To simplify this expression, we need to apply the rules of exponents.

Rule 1: Power of a Power

When we have a power raised to another power, we multiply the exponents. In this case, we have $\left(12 k^2 m^3 n^6\right)^2$ . To simplify this, we need to apply the rule of power of a power, which states that $(a^m)^n = a^{mn}$ .

Applying the Rule

Using the rule of power of a power, we can simplify the expression as follows:

$\left(12 k^2 m^3 n^6\right)^2 = (12)^2 (k^2)^2 (m^3)^2 (n^6)^2$

Simplifying Each Term

Now, we can simplify each term using the rule of power of a power:

$(12)^2 = 12^2 = 144$
$(k^2)^2 = k^{2 \cdot 2} = k^4$
$(m^3)^2 = m^{3 \cdot 2} = m^6$
$(n^6)^2 = n^{6 \cdot 2} = n^{12}$

Combining the Terms

Now, we can combine the simplified terms:

$144 k^4 m^6 n^{12}$

Conclusion

In this example, we used the rule of power of a power to simplify the expression $\left(12 k^2 m^3 n^6\right)^2$ . We applied the rule to each term and simplified the expression to $144 k^4 m^6 n^{12}$ .

Examples of Simplifying Expressions Using Exponents

Example 1: Simplifying $\left(3 x^2 y^3\right)^3$

To simplify this expression, we need to apply the rule of power of a power. We have a base of $3 x^2 y^3$ and an exponent of $3$ . Using the rule, we can simplify the expression as follows:

$\left(3 x^2 y^3\right)^3 = (3)^3 (x^2)^3 (y^3)^3$

Simplifying each term, we get:

$(3)^3 = 3^3 = 27$
$(x^2)^3 = x^{2 \cdot 3} = x^6$
$(y^3)^3 = y^{3 \cdot 3} = y^9$

Combining the terms, we get:

$27 x^6 y^9$

Example 2: Simplifying $\left(2 a^3 b^2\right)^4$

To simplify this expression, we need to apply the rule of power of a power. We have a base of $2 a^3 b^2$ and an exponent of $4$ . Using the rule, we can simplify the expression as follows:

$\left(2 a^3 b^2\right)^4 = (2)^4 (a^3)^4 (b^2)^4$

Simplifying each term, we get:

$(2)^4 = 2^4 = 16$
$(a^3)^4 = a^{3 \cdot 4} = a^{12}$
$(b^2)^4 = b^{2 \cdot 4} = b^8$

Combining the terms, we get:

$16 a^{12} b^8$

Tips for Simplifying Expressions Using Exponents

Make sure to apply the rule of power of a power correctly.
Simplify each term separately before combining them.
Use the correct order of operations (PEMDAS) when simplifying expressions.

Conclusion

Simplifying expressions using exponents can be a challenging task, but with practice and patience, you can master it. Remember to apply the rule of power of a power correctly, simplify each term separately, and use the correct order of operations. With these tips and examples, you'll be able to simplify expressions using exponents like a pro!

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Frequently Asked Questions

Q: What is the rule of power of a power?

A: The rule of power of a power states that $(a^m)^n = a^{mn}$ . This means that when we have a power raised to another power, we multiply the exponents.

Q: How do I simplify an expression using the rule of power of a power?

A: To simplify an expression using the rule of power of a power, you need to apply the rule to each term in the expression. This involves multiplying the exponents of each term.

Q: What is the order of operations when simplifying expressions using exponents?

A: The order of operations when simplifying expressions using exponents is:

Apply the rule of power of a power.
Simplify each term separately.
Combine the simplified terms.

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

A: To simplify an expression with multiple terms, you need to apply the rule of power of a power to each term separately. Then, combine the simplified terms.

Q: What is the difference between a power and an exponent?

A: A power is the base number that is being raised to a certain power, while an exponent is the number that the base is being raised to.

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

A: To simplify an expression with a negative exponent, you need to apply the rule of negative exponents, which states that $a^{-m} = \frac{1}{a^m}$ .

Q: Can I simplify an expression with a variable in the exponent?

A: Yes, you can simplify an expression with a variable in the exponent. However, you need to apply the rule of power of a power correctly.

Q: How do I simplify an expression with multiple variables in the exponent?

A: To simplify an expression with multiple variables in the exponent, you need to apply the rule of power of a power to each variable separately. Then, combine the simplified terms.

Examples of Simplifying Expressions Using Exponents: Q&A

Q: Simplify $\left(3 x^2 y^3\right)^3$

A: To simplify this expression, we need to apply the rule of power of a power. We have a base of $3 x^2 y^3$ and an exponent of $3$ . Using the rule, we can simplify the expression as follows:

$\left(3 x^2 y^3\right)^3 = (3)^3 (x^2)^3 (y^3)^3$

Simplifying each term, we get:

$(3)^3 = 3^3 = 27$
$(x^2)^3 = x^{2 \cdot 3} = x^6$
$(y^3)^3 = y^{3 \cdot 3} = y^9$

Combining the terms, we get:

$27 x^6 y^9$

Q: Simplify $\left(2 a^3 b^2\right)^4$

A: To simplify this expression, we need to apply the rule of power of a power. We have a base of $2 a^3 b^2$ and an exponent of $4$ . Using the rule, we can simplify the expression as follows:

$\left(2 a^3 b^2\right)^4 = (2)^4 (a^3)^4 (b^2)^4$

Simplifying each term, we get:

$(2)^4 = 2^4 = 16$
$(a^3)^4 = a^{3 \cdot 4} = a^{12}$
$(b^2)^4 = b^{2 \cdot 4} = b^8$

Combining the terms, we get:

$16 a^{12} b^8$

Tips for Simplifying Expressions Using Exponents: Q&A

Make sure to apply the rule of power of a power correctly.
Simplify each term separately before combining them.
Use the correct order of operations (PEMDAS) when simplifying expressions.