Problem 76 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{8 a^{5}}\left(\sqrt{2 a}-\sqrt{6 a^{11}}\right) $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: \(2\sqrt{8a^5}\left(\sqrt{2a}-\sqrt{6a^{11}}\right)\) Answer: \(4a^4 - 4a^{13}\)

Step 1: Break down the square roots

Recall the property \(\sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}\). Using this, we can break down the given square roots into simpler expressions. $$ \sqrt{8 a^{5}} = \sqrt{2^3\cdot a\cdot a^4}=\sqrt{2^3}\cdot \sqrt{a}\cdot \sqrt{a^4} = 2\sqrt{2} a^2 \sqrt{a} $$ Now we can rewrite the given expression: $$ 2\sqrt{2} a^2 \sqrt{a}\left(\sqrt{2 a}-\sqrt{6 a^{11}}\right) $$

Step 2: Distributive property

Apply the distributive property to multiply the terms inside the parenthesis. $$ \begin{aligned} 2\sqrt{2} a^2 \sqrt{a}\left(\sqrt{2 a}-\sqrt{6 a^{11}}\right) &= 2\sqrt{2} a^2 \sqrt{a}\cdot\sqrt{2 a} - 2\sqrt{2} a^2 \sqrt{a}\cdot\sqrt{6 a^{11}} \end{aligned} $$

Step 3: Combine and simplify the terms

Now we can combine and simplify the terms obtained in the previous step. $$ \begin{aligned} 2\sqrt{2} a^2 \sqrt{a}\cdot\sqrt{2 a} &= 2\sqrt{2}\cdot a^2\cdot \sqrt{a}\cdot\sqrt{2}\cdot \sqrt{a} \\ &= 2\sqrt{2}\cdot 2 \cdot a^2 \cdot a \cdot \sqrt{a^2} \\\ &= 4 a^3 a = 4 a^4 \end{aligned} $$ $$ \begin{aligned} 2\sqrt{2} a^2 \sqrt{a}\cdot\sqrt{6 a^{11}} &= 2\sqrt{2}\cdot a^2 \cdot \sqrt{a}\cdot\sqrt{6}\cdot\sqrt{a^{11}} \\ &= 2\sqrt{2}\cdot \sqrt{6}\cdot a^2 \cdot a^{11} \cdot \sqrt{a^2} \\ &= 2\sqrt{12}\cdot a^{13} \\\ &= 4 a^{13} \end{aligned} $$ Now, put the simplified terms back together: $$ 4 a^4 - 4 a^{13} $$ The simplified expression is \(4 a^4 - 4 a^{13}\).

Key Concepts

Distributive Property

In algebra, the distributive property is an essential concept that helps simplify expressions by removing brackets. It tells us that a term outside the parentheses can be "distributed" to each term inside the parentheses. Using this method makes multiplication easier. Consider the formula:

• If you have an expression like \(a(b + c)\), it becomes \(ab + ac\) once the distributive property is applied.

In our exercise, this concept is applied to the expression \(2\sqrt{2} a^2 \sqrt{a}\left(\sqrt{2 a}-\sqrt{6 a^{11}}\right)\). Here, \(2\sqrt{2} a^2 \sqrt{a}\) is distributed to both \(\sqrt{2 a}\) and \(\sqrt{6 a^{11}}\), resulting in two separate products that are simplified individually. Breaking down expressions using this property can make complex algebraic manipulations much more manageable.

Square Roots

Square roots might seem tricky initially, but they are an important part of algebra. A square root is a value that, when multiplied by itself, gives the original number. Here’s the notation: if \(b^2 = a\), then \(b = \sqrt{a}\). Often, you'll see square roots combined with other algebraic operations.

### Breaking Down Square RootsSometimes, square roots are part of larger expressions, and we may need to split them into simpler terms. For this, we can use the property that \(\sqrt{a\cdot b} = \sqrt{a}\cdot \sqrt{b}\). For example, in the given problem, \(\sqrt{8a^5}\) is broken down into \(\sqrt{2^3}\cdot \sqrt{a^4}\cdot \sqrt{a}\), simplified further as \(2\sqrt{2} a^2 \sqrt{a}\). This method helps simplify and make calculations more manageable, especially when the square root is part of a larger algebraic fraction.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. The goal is to rewrite expressions in a simpler or more compact form. Let's look at how we can achieve this.

### Combining Like TermsOne of the first things to do is combine like terms—terms that have the same variables raised to the same power. This helps reduce complexity. For example, \(2x + 3x\) simplifies to \(5x\).

### Handling Square RootsSimplifying expressions with square roots often involves breaking down the components first, as demonstrated in the problem when breaking up \(\sqrt{8a^5}\) into easily manageable terms.

### Final ReviewOnce you've simplified as far as you can, double-check it. For instance, after applying properties and combining terms, our example results in the final simplified expression: \(4a^4 - 4a^{13}\). Here, no further simplification is possible because the terms involve different powers of \(a\). This final check ensures that your expression is in its simplest form, making it easier to understand or use in further calculations.