Problem 58 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ 4 a x^{2} \sqrt{75 x^{4}}+6 a \sqrt{3 x^{8}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(26a x^{4} \sqrt{3}\).

Step 1: Identify like terms

First, identify like terms in the expression: \(4 a x^{2} \sqrt{75 x^{4}}\) and \(6 a \sqrt{3 x^{8}}\). Both terms have the variable 'a', square roots, and the variables 'x' within the square roots.

Step 2: Simplify square roots

Simplify each square root individually. For \(\sqrt{75 x^{4}}\): 1. Break down 75: \(75 = 5^2 \cdot 3\). 2. Factor out perfect squares (pairs of numbers): \(5^2\) in this case. 3. Simplify the expression considering \(x^{4}\): \(\sqrt{75 x^{4}} = \sqrt{3 \cdot 25 x^{4}} = 5x^2\sqrt{3}\) For \(\sqrt{3x^{8}}\): 1. Simplify the expression by considering \(x^{8}\) as: \(\sqrt{3 x^{8}} = \sqrt{3}\cdot x^{4}\)

Step 3: Multiply the terms with the simplified square roots

Now, multiply the simplified square roots with the rest of the expression. For the first term: \(4 a x^{2} (5 x^{2} \sqrt{3}) = 20 a x^{4} \sqrt{3}\) For the second term: \(6 a (\sqrt{3} x^{4}) = 6 a x^{4} \sqrt{3}\)

Step 4: Combine like terms

Combine the like terms from step 3. \(20 a x^{4} \sqrt{3} + 6 a x^{4} \sqrt{3} = (20+6) a x^{4} \sqrt{3}\)

Step 5: Simplify the expression

Finally, simplify the expression by performing the addition. \((20+6) a x^{4} \sqrt{3} = 26a x^{4} \sqrt{3}\) The simplified expression is: $$ 26a x^{4} \sqrt{3} $$

Key Concepts

Understanding Square Roots

Square roots are fascinating components of mathematics. They help us express a number that, when multiplied by itself, gives us the original number. In algebra, square roots often appear within expressions, as seen in our exercise problem. Handling these requires breaking down the components inside the square root. Let's consider an example. The square root of 75, written as \(\sqrt{75}\), can be simplified. We break 75 into its prime factors: \(75 = 5^2 \times 3\). The \(5^2\) is a perfect square, which can be brought out of the square root, resulting in \(5\sqrt{3}\). Simpler square roots make algebraic expressions easier to manage. When variables like \(x^4\) are present within square roots, we apply similar principles. Because \(x^4\) is a perfect square, representing \((x^2)^2\), it also comes out from the square root, leaving behind \(x^2\). Combining all these, \(\sqrt{75 x^{4}} = 5x^2\sqrt{3}\). By consistently applying these techniques, we can easily tackle even more complex algebraic expressions.

Combining Like Terms

In algebra, combining like terms is a crucial skill that simplifies expressions by consolidating similar components. Like terms are expressions that share the same variables and exponents. For example, in our problem, the terms \(20 a x^4 \sqrt{3}\) and \(6 a x^4 \sqrt{3}\) are considered like terms because they both contain \(a x^4 \sqrt{3}\). Recognizing like terms helps in streamlining expressions.Combining like terms involves summing their coefficients while maintaining the shared variable portion. In this context, coefficients are numerical factors placed before variables. Thus, when we add \(20 a x^4 \sqrt{3}\) and \(6 a x^4 \sqrt{3}\), we sum up the numbers - 20 and 6 - resulting in \(26 a x^4 \sqrt{3}\). This process simplifies the expression significantly.
Consistency in recognizing and combining like terms streamlines problem-solving and aids in clearer, more manageable algebraic expressions.

Recognizing Perfect Squares

Perfect squares are expressions obtained when an integer or a variable is squared. Their identification simplifies algebraic expressions, making manipulation straightforward. When simplifying square roots, as seen in the task, recognizing perfect squares inside the square root can drastically reduce the complexity of the expression.For example, \(75\) inside \(\sqrt{75}\) can be factored as \(5^2 \times 3\). Here, \(5^2\) is a perfect square. Pulling out the perfect square from under the root gives us \(5\sqrt{3}\). This process demonstrates the role perfect squares play in simplifying problems and highlights the ease with which square roots can be manipulated.Similarly, \(x^4\) is a perfect square because it equals \((x^2)^2\). Understanding and identifying perfect squares cut down calculations and allow for more direct simplification of expressions.

Simplifying Polynomial Expressions

Polynomial expressions are algebraic expressions composed of variables and coefficients, linked by addition, subtraction, multiplication, and non-negative integer exponents. Our original expression \(4 a x^2 \sqrt{75 x^4} + 6 a \sqrt{3 x^8}\) is a polynomial comprising two parts. Simplification involves addressing each componentFirst, simplifying the square roots makes the polynomial terms easier to handle. By breaking \(75 x^4\) and \(3 x^8\) into their factors, we simplify to expressions like \(5 x^2 \sqrt{3}\) and \(x^4 \sqrt{3}\), respectively. The next step involves multiplying these simplified forms back into their original polynomial parts. After this, the expression becomes \(20 a x^4 \sqrt{3} + 6 a x^4 \sqrt{3}\). Finally, by combining like terms, we reduce the polynomial into its simplest form: \(26 a x^4 \sqrt{3}\). Such streamlined simplification enhances the understanding and handling of polynomial expressions.