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Chapter 10: Problem 47

Simplify. See Examples 3 and 4 $$ \sqrt{25 a^{2} b^{3}} $$

Short Answer

Expert verified
The simplified form is \( 5ab\sqrt{b} \).

Step by step solution

01

Separate the components under the square root

The expression under the square root is \( 25a^2b^3 \). We will break it down into separate components: \( \sqrt{25} \), \( \sqrt{a^2} \), and \( \sqrt{b^3} \).
02

Simplify each component

Now, we simplify each component separately. \( \sqrt{25} = 5 \), \( \sqrt{a^2} = a \), and \( \sqrt{b^3} = b\sqrt{b} \).
03

Combine simplified components

Combine the simplified components from Step 2. The result is \( 5ab\sqrt{b} \).

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Key Concepts

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

Understanding Square Roots
Square roots represent a special number that, when multiplied by itself, gives the original number. In algebra, square roots allow us to simplify expressions by breaking down products into perfect squares. For instance, the square root of 25 is 5, because 5 times 5 equals 25.

When working with variables, square roots follow similar rules. Take \( a^2 \), for instance. The square root of \( a^2 \) is \( a \), because \( a \times a = a^2 \). Similarly, when dealing with a square root like \( b^3 \), you think of it as \( \sqrt{b^2\cdot b} = b\sqrt{b} \).

Remember:
  • Simplify square roots by finding perfect squares.
  • Break down variables according to their exponents.
  • Look for patterns and repeated values to ease calculations.
Demystifying Exponents
Exponents are a fundamental concept in algebra that represent repeated multiplication. For example, \( 3^2 \) means 3 multiplied by itself, equaling 9. This concept extends to any number or variable with an exponent. Simplifying expressions involves understanding how these powers work.

When you encounter variables with exponents under a square root, like \( a^2 \) or \( b^3 \), it's important to break them down. \( a^2 \) is straightforward; its square root is simply \( a \). But with \( b^3 \), you see it as \( b^2 \cdot b \). So, the square root becomes \( b\sqrt{b} \), because you can only fully "extract" the perfect square part.

Key points:
  • Exponents stand for multiplication patterns.
  • When simplifying, divide exponents by 2 under a square root.
  • Leftover parts stay within the square root in their most reduced form.
Simplifying Expressions
Simplifying expressions in algebra involves reducing them to their simplest form. This might include removing parentheses, factoring, or simplifying intricate terms like square roots or powers. This process helps in making math expressions more manageable and interpretable.

Consider the expression \( \sqrt{25a^2b^3} \). First, break it into \( \sqrt{25} \), \( \sqrt{a^2} \), and \( \sqrt{b^3} \). By simplifying these separately, you turn \( \sqrt{25} \) into 5, \( \sqrt{a^2} \) into \( a \), and \( \sqrt{b^3} \) into \( b\sqrt{b} \). Joining these together gives \( 5ab\sqrt{b} \).

Tips for simplifying:
  • Work on each component individually.
  • Identify and simplify perfect squares or powers.
  • Recombine the simplified parts logically.

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Most popular questions from this chapter

Identify the domain and then graph each function. $$g(x)=\sqrt[3]{x-1}$$ use the following table. $$\begin{array}{|c|c|} \hline x & {g(x)} \\ \hline 1 & {} \\ \hline 2 & {} \\ \hline 0 & {} \\ \hline 9 \\ \hline-7 \\ \hline \end{array}$$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$f(-1)$$

Use rational expressions to write as a single radical expression. $$ \sqrt[4]{5} \cdot \sqrt[3]{x} $$

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{5} \cdot \sqrt{2} $$

Simplify each exponential expression. $$\left(-14 a^{5} b c^{2}\right)\left(2 a b c^{4}\right)$$
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