Problem 43 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \frac{\sqrt{45 a^{3} b^{8} c^{2}}}{\sqrt{5 a b^{2} c}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{\sqrt{45 a^{3} b^{8} c^{2}}}{\sqrt{5 a b^{2} c}}\). Answer: \(3ab^{3}\sqrt{c}\)

Step 1: Rewrite the expression with a single square root

Use the property of square roots that states \(\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}\), we can rewrite the given expression as a single square root: $$ \frac{\sqrt{45 a^{3} b^{8} c^{2}}}{\sqrt{5 a b^{2} c}} = \sqrt{\frac{45 a^{3} b^{8} c^{2}}{5 a b^{2} c}} $$

Step 2: Simplify the terms inside the square root

Simplify the expression inside the square root by dividing each term in the numerator by the corresponding term in the denominator: $$ \sqrt{\frac{45 a^{3} b^{8} c^{2}}{5 a b^{2} c}} = \sqrt{\frac{45}{5} \cdot \frac{a^{3}}{a} \cdot \frac{b^{8}}{b^{2}} \cdot \frac{c^{2}}{c}} $$

Step 3:Perform the divisions inside the square root

Now, we will perform the divisions inside the square root: $$ \sqrt{\frac{45}{5} \cdot \frac{a^{3}}{a} \cdot \frac{b^{8}}{b^{2}} \cdot \frac{c^{2}}{c}} = \sqrt{9 \cdot a^{2} \cdot b^{6} \cdot c} $$

Step 4: Find the square root

Finally, we will take the square root of the simplified expression inside the square root: $$ \sqrt{9 \cdot a^{2} \cdot b^{6} \cdot c} = 3ab^{3}\sqrt{c} $$ So the simplified expression is \(3ab^{3}\sqrt{c}\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is a process that involves reducing an expression to its most concise and efficient form. It requires combining like terms, using mathematical properties, and sometimes applying factorization. Let’s break down how it works using an example from a square root problem.

Consider this expression inside a square root:

• First, identify terms that can be grouped or reduced. In this case, terms like \( \frac{45a^3b^8c^2}{5ab^2c} \) can be simplified.
  
• Divide each of the terms in the numerator by their corresponding terms in the denominator. This involves basic division: \( \frac{45}{5} = 9 \) and similarly for the variables resulting in \( a^2 \), \( b^6 \), and \( c \).
  
• Simplifying makes it easier to handle expressions and solve equations, allowing for clearer understanding and computation.

By following these steps, you ensure expressions are manageable and less error-prone when dealing with complex equations.

Square Roots

Square roots help in finding a number which, when multiplied by itself, gives the original number. The symbol \( \sqrt{} \), denotes a square root, and it's pivotal in simplifying radical expressions.

To simplify the square root in our example:

• Combine the entire fraction under one square root initially: \( \sqrt{\frac{45a^3b^8c^2}{5ab^2c}} \), instead of keeping separate terms. This is a crucial first step, streamlining the simplification process.
  
• Identify perfect squares within the fraction, such as \( 9 \) – the square of \( 3 \), and other perfect power combinations of variables.
  
• Finally, take the square root separately for each factor: the constant \( 9 \), and variables \( a^2 \) and \( b^6 \), resulting in \( 3ab^3 \) outside the root.
  

Leveraging the properties of square roots saves effort and guides you toward the cleanest answer, as seen in obtaining \( 3ab^3\sqrt{c} \).

Exponents

Exponents are essential in algebra, providing a way to express repeated multiplication of a number by itself succinctly. Understanding the rules of exponents is key to efficiently solving problems involving powers.

In our problem:

• Recognize that when dividing terms with the same base, you subtract the exponents: \( \frac{a^3}{a} = a^{3-1} = a^2 \). This same rule applies to all like-variable terms.
• Another rule involves multiplication: adding the exponents if multiplying similar bases, although this was not required here, it applies broadly in solving such expressions.
  
• Lastly, the power of zero and the negative exponent rules are helpful tools in broader contexts, offering a complete set of strategies to manipulate algebraic expressions.
  

Mastering exponents and their properties ensures you can handle a wide range of mathematical problems with confidence and precision.