Intro to the Power Rules: Videos & Practice Problems

Understanding exponential expressions involves applying rules like the power rule, where an exponential expression raised to another exponent simplifies by multiplying the exponents, e.g., $a^{m\times n}$. The power of a product rule distributes an exponent across factors in a product, such as ${x\times y}^n=x^n\times y^n$. These rules simplify polynomials and multivariable expressions, enhancing skills in manipulating terms, coefficients, and powers efficiently.

The Power Rule for Exponents

When working with exponential expressions, understanding how to simplify powers raised to other powers is essential. If you have an expression like four cubed raised to the second power, this means you are multiplying the base raised to an exponent by itself multiple times. Specifically, (4³)² can be rewritten as 4³ × 4³. Since the bases are the same, you apply the product rule of exponents, which states that when multiplying like bases, you add the exponents. So, 3 + 3 = 6, simplifying the expression to 4⁶.

Alternatively, you can use the power rule for exponents, which states that when an exponential expression is raised to another power, you multiply the exponents. This means (4³)² = 4^{3 × 2} = 4⁶. This method is especially useful for larger exponents because multiplication is a more efficient operation than repeated addition.

For example, consider (-2³)⁵. Applying the power rule, multiply the exponents: 3 × 5 = 15, so the expression simplifies to (-2)^{15}. Evaluating this gives -32768. Similarly, for (y⁸)⁴, multiply the exponents to get y^{8 × 4} = y^{32}.

In summary, the power rule for exponents is a fundamental tool for simplifying expressions where a power is raised to another power. It states that (a^m)^n = a^{m × n}, where a is the base and m and n are exponents. This rule streamlines calculations and deepens understanding of exponential relationships.

Simplify each expression, but don’t evaluate.
${\left(4^{15}\right)}^3$

• A. $4^{18}$
• B. $4^5$
• C. $4^{12}$
• D. $4^{45}$

Answer: (D)

Simplify each expression, but don’t evaluate.
${\left({100}^{26}\right)}^0$

• A. $0$
• B. $100^{26}$
• C. $100^0$
• D. $100^{\infty}$

Answer: (C)

Simplify each expression, but don’t evaluate.
$\frac{(x^2{)}^4}{{\left(x^3\right)}^3}$

• A. $x^{\frac89}$
• B. $x^{-1}$
• C. $x^1$
• D. $x^{\frac12}$

Answer: (B)

The Power Rule for Exponents Example 1

When evaluating expressions involving powers raised to another power, such as 3⁴² and 3²⁴, it is important to apply the power of a power rule. This rule states that when an exponent is raised to another exponent, you multiply the exponents together. For example, (a^m)ⁿ = a^{m × n}.

Applying this to the expressions, 3⁴² can be rewritten as 3^{4 × 2} = 3⁸, and similarly, 3²⁴ becomes 3^{2 × 4} = 3⁸. Since multiplication of exponents is commutative, both expressions simplify to the same value, which is 3⁸.

Calculating 3⁸ yields 6,561, confirming that both expressions are equal. This demonstrates the importance of understanding exponent rules to simplify and compare expressions efficiently.

The Power Rule for Exponents Example 2

When working with exponents, the power rule is a fundamental concept that states if you have an exponent raised to another exponent, you multiply the exponents to simplify the expression. For example, if you have an expression like \(x^a\) raised to the power of \(b\), it simplifies to \(x^{a \times b}\). This means that to find equivalent expressions for \(x^{15}\), you can look for pairs of numbers whose product is 15.

Since 15 can be factored into several pairs such as 1 and 15, 15 and 1, 3 and 5, or 5 and 3, each pair can represent the exponents in the expression \((x^a)^b\). For instance, \((x^1)^{15}\) simplifies to \(x^{1 \times 15} = x^{15}\), and similarly, \((x^{15})^1\) also equals \(x^{15}\). Likewise, \((x^3)^5\) and \((x^5)^3\) both simplify to \(x^{15}\) because \$3 \times 5 = 15\( and \)5 \times 3 = 15$ respectively.

This understanding of the power rule not only helps in simplifying expressions but also in recognizing equivalent forms of exponential expressions by factoring the exponent into different pairs of integers. It reinforces the importance of multiplication in exponentiation and provides flexibility in manipulating powers for algebraic simplification.

Power of a Product Rule

The power of a product rule is a fundamental exponent rule that allows you to simplify expressions where a product of two or more factors is raised to a single exponent. This rule states that when you have a product inside parentheses raised to an exponent, you can distribute the exponent to each factor within the product. For example, the expression \((3 \times 4)^2\) can be rewritten by applying the exponent to both 3 and 4 individually, resulting in \$3^2 \times 4^2\(. This works because squaring the product means multiplying the entire quantity by itself, which is equivalent to multiplying each factor by itself.

Understanding this rule helps in simplifying complex exponential expressions efficiently. For instance, consider the expression \)(5 \times 3^2)^2\(. By distributing the outer exponent, it becomes \)5^2 \times (3^2)^2\(. Then, using the power of a power rule, which states that \)(a^m)^n = a^{m \times n}\(, the expression simplifies further to \)5^2 \times 3^{2 \times 2} = 5^2 \times 3^4\(. Evaluating this yields \)25 \times 81 = 2025\(.

Another example involves variables: \)(xy)^5\(. Since variables placed side by side imply multiplication, the power of a product rule allows us to rewrite this as \)x^5 \times y^5$. This simplification is crucial for working with algebraic expressions involving exponents.

In summary, the power of a product rule is applied whenever a product is raised to an exponent. It enables the distribution of the exponent to each factor, simplifying the expression and making it easier to evaluate or manipulate further. This rule, combined with other exponent rules like the power of a power rule, forms the foundation for working with exponential expressions in algebra.

Simplify each expression.
${\left(4x^2\right)}^3$

• A. $4x^6$
• B. $12x^5$
• C. $64x^6$
• D. $64x^5$

Answer: (C)

Simplify each expression.
$(-6^5{)}^{{}^5}$

• A. $\left(-6\right)^{25}$
• B. $\left(-6\right)^{10}$
• C. $6^{25}$
• D. $\left(-6\right)^5$

Answer: (A)

Simplify each expression.
$(x^2{z}^5{)}^4$

• A. $x^6z^9$
• B. $x^{20}z^8$
• C. $x^9z^6$
• D. $x^8z^{20}$

Answer: (D)

Simplify each expression.
$(-2x^4{y}^5{)}^4$

• A. $-16x^{16}y^{20}$
• B. $2x^{16}y^{20}$
• C. $16x^{16}y^{20}$
• D. $8x^8y^9$

Answer: (C)

Simplify each expression.
$(4x^3{y}^2{z}^3{)}^2$

• A. $8x^6y^4z^6$
• B. $16x^6y^4z^6$
• C. $16x^3y^2z^3$
• D. $4x^3y^4z^6$

Answer: (B)

Here’s what students ask on this topic:

What is the power rule in exponents and how do you apply it?

The power rule in exponents states that when you have an exponential expression raised to another exponent, you multiply the exponents. For example, if you have $a^m$ raised to the power $n$, it simplifies to $a^{m\times n}$. This means ${a^m}^n=a^{m\times n}$. For example, ${4^3}^2=4^6$ because 3 times 2 equals 6. This rule helps simplify expressions quickly without expanding them fully, especially useful for large exponents.

How does the power of a product rule work in simplifying exponential expressions?

The power of a product rule allows you to distribute an exponent across factors inside parentheses. If you have a product like $(x\times y)n^{}$, you can rewrite it as $x^n\times y^n$. For example, ${(}^3=3^2\times 4^2$. This simplifies to $9\times 16=144$. This rule is useful for breaking down complex expressions into simpler parts by applying the exponent to each factor individually.

Can you explain how to simplify an expression like (x³)⁴ using the power rule?

To simplify ${x^3}^4$, apply the power rule by multiplying the exponents. Multiply 3 by 4 to get 12, so the expression becomes $x^{12}$. This means ${x^3}^4=x^{12}$. This method is much faster than expanding the expression as $x^3\times x^3\times x^3\times x^3$ and then adding exponents repeatedly.

What is the difference between the product rule and the power of a product rule in exponents?

The product rule applies when multiplying exponential expressions with the same base, such as $a^m\times a^n=a^{m+n}$. You add the exponents because the bases are the same. The power of a product rule applies when a product inside parentheses is raised to an exponent, like ${(}^a=a^n\times b^n$. Here, you distribute the exponent to each factor. The key difference is that the product rule deals with multiplying powers of the same base, while the power of a product rule distributes an exponent over a product of different bases.

How do you simplify an expression like (5 × 3²)² using the power of a product rule?

To simplify ${(}^5$, first distribute the exponent 2 to each factor inside the parentheses using the power of a product rule. This gives $5^2\times {3^2}^2$. Then apply the power rule to the second term by multiplying the exponents: 2 × 2 = 4, so it becomes $5^2\times 3^4$. Finally, calculate the values: $25\times 81=2025$. This method simplifies the expression efficiently without expanding all terms manually.