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Chapter 7: Problem 41

Simplify the products. Give exact answers. $$(2 \sqrt{3})^{2}$$

Short Answer

Expert verified
12

Step by step solution

01

- Apply the Exponent

First, remember that applying the exponent to a product means applying it to each factor in the product. In this case, \( (2 \sqrt{3})^{2} = 2^{2} \cdot (\sqrt{3})^{2} \).
02

- Simplify Each Factor

Now, simplify each part separately. For \(2^{2}\), we have \(2 \times 2 = 4\). For \((\sqrt{3})^{2}\), recall that \(\sqrt{a}^{2} = a\), so it simplifies to \(3\).
03

- Multiply the Results

Finally, multiply the simplified results together: \(4 \cdot 3 = 12\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Exponents
Exponents are a shorthand way to denote repeated multiplication of the same number. In our exercise, we encounter an exponent of 2. When we have an expression like \(a^b\), it means multiplying \(a\) by itself \(b\) times. So, \(2^2\) simplifies to \(2 \times 2 = 4\).

Remember, exponents apply to the entire term inside the parentheses. Therefore, when dealing with a product inside parentheses, each factor gets raised to the exponent. In the case of \( (2 \sqrt{3})^2 \), we apply the exponent 2 to both \( 2 \) and \( \sqrt{3} \).

By breaking it down, we get:
  • \(2^2 \) simplifies to 4, which is directly found using multiplication.
  • \((\sqrt{3})^2\) simplifies to 3, which we explore in more detail in the next section.
Simplifying Square Roots
A square root is a value that, when multiplied by itself, gives the original number. For example, \sqrt{3} \ times \sqrt{3} = 3 \. This idea helps us simplify expressions with square roots raised to an exponent.

When you see \((\sqrt{a})^2\), it simplifies directly to \(a\), because \(\sqrt{a} \times \sqrt{a} \ = a \). This neat property helps greatly in our problem.

In our exercise, \((\sqrt{3})^2\) simplifies directly to \3 \. By knowing this simplification, you can easily handle problems involving square roots and exponents.
  • To sum up, whenever you encounter a square root raised to an exponent of 2, you can directly replace it with the number inside the root.
Calculating Products
Products in mathematics refer to the result of multiplying numbers together. After applying exponents and simplifying each factor separately, you multiply the simplified results.

In our example, we simplify \(2^2 = 4\) and \((\sqrt{3})^2 = 3\). The final step involves multiplying these two simplified values together to get the final product.

\(4 \times 3 = 12 \). This multiplication gives us 12, which is the exact answer to our original problem \((2 \sqrt{3})^2 \).

Remember, when simplifying expressions, always break them down into individual parts, apply exponents, and then multiply the results.
  • Start with applying exponents.
  • Simplify individually, e.g., square roots and other factors.
  • Multiply the simplified results.

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Most popular questions from this chapter

Solve each equation. $$(3 a-1)^{-2 / 5}=1$$

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