Question

Which expression is equivalent to \(4^6 \cdot 2^6\) ? A. \(6^{36}\) B. \(8^{12}\) C. \(6^{12}\) D. \(8^6\)

Short answer

The expression equivalent to \4^6 \cdot 2^6\ is \ 8^6 \.

Step-by-step solution

- Analyze the Given Expression

Start by looking at the given expression: \(4^6 \cdot 2^6\).

- Rewrite the Base in Terms of the Same Number

Observe that we can express 4 as a power of 2, since \(4 = 2^2\). Therefore, rewrite the expression as: \(4^6 = (2^2)^6\).

- Apply the Power Rule

Use the power rule \((a^m)^n = a^{m \cdot n}\)\ to simplify \((2^2)^6\) further: \((2^2)^6 = 2^{2 \cdot 6} = 2^{12}\).

- Combine the Expression

Now, simplify the total expression: \(2^{12} \cdot 2^6\). Since the bases are the same, add the exponents: \( 2^{12+6} = 2^{18} \).

- Match with the Given Choices

Review the choices: A. \ 6^{36} \ B. \ 8^{12} \ C. \ 6^{12} \ D. \ 8^6 \ Simplify \(8^6\): since \ 8 = 2^3 \, \ (2^3)^6 = 2^{3 \cdot 6} = 2^{18}\ The equivalent expression is \ 8^6 \.

Key concepts

Exponent Rules

Understanding exponent rules is crucial when simplifying expressions. In our exercise, we start with exponents, which involve multiplying powers of numbers. There are several important exponent rules:

• The Product Rule: When you multiply like bases, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• The Power Rule: When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).
• The Quotient Rule: When dividing like bases, you subtract the exponents: \(a^m / a^n = a^{m-n}\).

An understanding of these rules helps us simplify the expression \(4^6 \times 2^6\). Use the power rule to rewrite \(4^6\) in terms of base 2. Then, use the product rule to combine the exponents.

Power Rule

The power rule is particularly useful when dealing with expressions like our given problem. The power rule states \((a^m)^n = a^{m \times n}\). In this exercise, we applied the power rule to convert \(4^6\) by recognizing that 4 can be rewritten as \(2^2\). Thus, \(4^6 = (2^2)^6\).

• First, rewrite the base in terms of the same number: \4 = 2^2 \.
• Next, apply the power rule: \((2^2)^6 = 2^{2 \times 6} = 2^{12}\).

Once \(4^6\) is simplified to \(2^{12}\), we combine it with \(2^6\) using the product rule. This step is essential for further simplifying the expression.

Combining Like Terms

Combining like terms in expressions with exponents means adding exponents of the same base. In the given problem, we have already simplified \(4^6 \) as \2^{12}\:. Now, we combine \2^{12}\ and \2^6\:

• Identify the matching bases: Here, both terms have base 2.
• Apply the product rule: \2^{12} \times 2^6 = 2^{12+6} = 2^{18}\.

Finally, we compare the simplified expression \2^{18}\ to the given answer choices and find that \8^6\ simplifies to \2^{18}\, making it the correct match. This approach demonstrates how combining like terms simplifies expressions and makes it easier to match them with the given choices.