Problem 140 - Elementary and Intermediate Algebra

Question

Simplify \(\sqrt{9 a^{16}+12 a^{8} b^{25}+4 b^{50}}\) and assume that \(a>0\) and \(b>0\)

Step-by-Step Solution

Answer

The simplified form is \(3a^8 + 2b^{25}\).

Step 1: Identify Complete Squares

Recognize that the expression inside the square root has the form of a perfect square. The expression is structured as \[( ext{expression})^2 = 9 a^{16} + 12 a^{8} b^{25} + 4 b^{50} \]Notice that each term is a square or can be related to a square.

Step 2: Break Down the Expression

Break down each term in the expression as follows:- \(9 a^{16} = (3a^8)^2\)- \(12 a^8 b^{25} = 2 \cdot 3a^8 \cdot 2b^{25}\)- \(4 b^{50} = (2b^{25})^2\)This corresponds to \[(3a^8 + 2b^{25})^2\].

Step 3: Verify the Perfect Square

Check if the expression inside the parenthesis, \(3a^8 + 2b^{25}\), when squared gives the original expression:\[(3a^8 + 2b^{25})^2 = (3a^8)^2 + 2 \cdot 3a^8 \cdot 2b^{25} + (2b^{25})^2 \]Re-calculate the expansion:- \((3a^8)^2 = 9a^{16}\)- \(2 \cdot 3a^8 \cdot 2b^{25} = 12a^8b^{25}\)- \((2b^{25})^2 = 4b^{50}\)This confirms that the expression is indeed \[(3a^8 + 2b^{25})^2\].

Step 4: Simplify the Square Root

With the expression verified as a perfect square, simplify the square root:\[\sqrt{(3a^8 + 2b^{25})^2} = 3a^8 + 2b^{25}\]The importance of \(a > 0\) and \(b > 0\) ensures that we do not need absolute values, making the simplification direct.

Key Concepts

perfect squares

Perfect squares are expressions that can be written as the square of another expression. Identifying a perfect square is crucial in simplifying square roots, as it allows us to simplify the expression directly. In our exercise, the expression inside the square root is originally:

• \[9 a^{16} + 12 a^8 b^{25} + 4 b^{50}\]

This is structured as a perfect square:

• \[(3a^8 + 2b^{25})^2\]

Perfect squares usually follow the formula

• \((A + B)^2 = A^2 + 2AB + B^2\)

In this case, \(A\) is \(3a^8\) and \(B\) is \(2b^{25}\). Recognizing these terms helps to rewrite the quadratic expression inside the square root as a single squared expression, simplifying the process of further calculations.
Understanding this concept makes it easier to take the square root effectively.

algebraic expressions

Algebraic expressions involve a combination of variables, constants, and operational symbols arranged according to mathematical laws. In our original exercise:

• \[9 a^{16} + 12 a^8 b^{25} + 4 b^{50}\]

The expression is composed of several terms, each with coefficients and products of variables raised to powers. Here, the variables \(a\) and \(b\) have specific roles depending on their exponents in each of the terms.
Breaking down complex algebraic expressions into recognizable forms such as perfect squares can greatly simplify problem-solving. This involves identifying like terms, factoring, and rewriting the expression into a more manageable form, such as

• \((3a^8 + 2b^{25})^2\)

By recognizing the structure of the expression, we directly simplify it by using our understanding of how algebraic identities work.

exponents and radicals

Exponents and radicals are fundamental concepts in algebra that describe repeated multiplication and roots. For exponents, the expression \(a^{16}\) indicates that the variable \(a\) is multiplied by itself sixteen times. Similarly, \(b^{50}\) means that \(b\) is multiplied fifty times.
When combining exponents, especially when creating or recognizing perfect squares, it's crucial to reorganize these powers to match the forms of common algebraic identities. Radicals, like square roots, undo these exponents by returning the base number raised to half the original power when perfect squares are involved.
In our expression, simplifying the square root \[\sqrt{9 a^{16} + 12 a^8 b^{25} + 4 b^{50}}\] involves recognizing that the expression is a perfect square,

• \[(3a^8 + 2b^{25})^2\]

and then taking the square root results directly in \(3a^8 + 2b^{25}\). This demonstrates how exponents and radicals interact, simplifying potentially complex expressions with the right recognition skills and techniques.