Question

Simplify \(2^3 \cdot 4^3 \cdot 5^3\). A. \(40^9\) B. \(11^{27}\) C. \(40^3\) D. \(11^9\)

Short answer

Option C: \(40^3\).

Step-by-step solution

- Simplify each term using exponents

First, recognize that each term is raised to the power of 3. So, we have \(2^3 = 8\), \(4^3 = 64\), and \(5^3 = 125\).

- Multiply the simplified terms together

Next, multiply the simplified terms together: \(8 \times 64 \times 125\).

- Express the product as a single exponent

Recognize that multiplying these terms that are each raised to the same power of 3 can be simplified by: \(8 \times 64 \times 125 = (2 \times 4 \times 5)^3\).

- Calculate the base

Calculate \(2 \times 4 \times 5\), which equals \(40\). Therefore, the expression is simplified to: \(40^3\).

- Identify the correct option

Compare the result with the given choices. The correct answer is: Option C: \(40^3\).

Key concepts

simplifying exponents

Understanding how to simplify exponents is essential when dealing with terms raised to the same power. In this exercise, we have three terms: \(2^3\), \(4^3\), and \(5^3\). Each term is raised to the power of 3, so we can first calculate each base raised to its exponent separately. Simplifying these individually, we get:

• \(2^3 = 8\)
• \(4^3 = 64\)
• \(5^3 = 125\)

Once we have simplified each term, our expression can now be rewritten in simplified terms as: \(8 \times 64 \times 125\).

multiplication of terms with exponents

When multiplying terms that are each raised to a power, there are a few steps to follow. After simplifying each term in the previous section, we end up with \(8 \times 64 \times 125\). The key here is to realize that these products can further be simplified by a common base concept.
Notice all terms have the same exponent (3 in this case), so we can rewrite the multiplication of these as the product of their bases raised to the same power: \(8 \times 64 \times 125\) is the same as \((2 \times 4 \times 5)^3\). This multiplication inside the parentheses is much simpler to handle.

calculating product of bases

After simplifying and reshaping our exponents, the next step is to multiply the bases themselves: \(2 \times 4 \times 5\). Performing the multiplication step-by-step:

• First, calculate \(2 \times 4 = 8\)
• Then, multiply the result by 5: \(8 \times 5 = 40\)

Now that we have the product of the bases, we can rewrite our expression with the base 40 raised to the 3rd power: \(40^3\). Therefore, the final simplified form is \(40^3\), which matches Option C from the list of choices.