Problem 80 - Elementary Algebra

Question

For the following problems, simplify each expression by removing the radical sign. $$ \sqrt{64 a^{10}(a+4)^{14}} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: \(\sqrt{64 a^{10}(a+4)^{14}}\) Answer: \(8 a^5 (a+4)^7\)

Step 1: Identify the constant and variables

We first need to identify the constant and the variables inside the square root: $$ \sqrt{64 a^{10}(a+4)^{14}} $$ We have a constant: \(64\), a power of a variable: \(a^{10}\), and a power of a sum: \((a+4)^{14}\).

Step 2: Apply the properties of square roots and simplification

We can use the property of square roots, \(\sqrt{ab} = \sqrt{a}\sqrt{b}\), to separate the constant and variables: $$ \sqrt{64} \sqrt{a^{10}} \sqrt{(a+4)^{14}} $$ Now, we can simplify each part: $$ 8 a^5 (a+4)^7 $$ So, the simplified expression without the radical sign is: $$ 8 a^5 (a+4)^7 $$

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, often encountered in algebra. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, in our problem, \( \sqrt{64} \), we look to find a number that, when squared, equals 64. This number is 8, since \( 8 \times 8 = 64 \). So, \( \sqrt{64} = 8 \).

When you deal with square roots in algebraic expressions, the same principle applies. You need to look for terms that can be "rooted" perfectly. The square root of an expression like \( \sqrt{a^{10}} \) becomes \( a^{5} \), because \( a^{5} \times a^{5} = a^{10} \).

• The expression inside the square root is often referred to as the 'radicand'.
• Understanding the principles of square roots allows us to simplify expressions by extracting perfect squares.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations combined in a mathematical phrase. They are foundational in solving equations and simplifying mathematical problems.

In the given exercise, the expression \( \sqrt{64 a^{10}(a+4)^{14}} \) is an example of an algebraic expression within a radical. By breaking down the expression:

• \( 64 \) is a constant.
• \( a^{10} \) involves powers and represents ten occurrences of the variable \( a \).
• \( (a+4)^{14} \) shows a sum that is raised to a power.

Understanding how to handle these elements is key to simplifying such expressions. Separating the terms based on operations like square roots aids in simplification. This process helps in reducing complexity and in calculating further in algebraic contexts.

Powers and Exponents

Powers and exponents are a way of expressing repeated multiplication in a compact form. For instance, \( a^{10} \) means \( a \) multiplied by itself 10 times. This succinct representation helps simplify expressions and calculations involving large numbers.

Exponents play a crucial role in simplifying the problem at hand. When you encounter an exponent under a square root, like \( \sqrt{a^{10}} \), you should recognize this as a scenario where exponents can be halved (if even), this turns the problem into a much simpler form: \( a^{5} \).

• The property \( \sqrt{a^b} = a^{b/2} \) is handy when dealing with squared bases.
• For \( (a+4)^{14} \), applying the principle results in \((a+4)^{7}\), which contributes to simplifying the large expression.

By mastering powers and exponents, you can break down and simplify complex algebraic expressions, making them more manageable for further calculations.