Problem 12 - Elementary and Intermediate Algebra

Question

Simplify: $$ \frac{\sqrt{80 s^{2} t^{4}}}{\sqrt{5 s^{2}}}=\sqrt{\frac{80 s^{2} t^{4}}{\underline{\phantom{xx}}}} $$ $$ \begin{aligned} &=\\\ &= \end{aligned} $$

Step-by-Step Solution

Answer

The simplified expression is \( 4t^2 \).

Step 1: Simplify the inside of the square root

First, simplify the expression inside the square root: \( \frac{80s^2t^4}{5s^2} \). Divide the coefficients and variables: \( \frac{80}{5} \) gives \( 16 \), and \( \frac{s^2}{s^2} \) is \( 1 \). Thus, the expression simplifies to \( 16t^4 \).

Step 2: Apply the square root

Now take the square root of the simplified expression \( \sqrt{16t^4} \). This can be broken down into \( \sqrt{16} \times \sqrt{t^4} \).

Step 3: Simplify individual square roots

Compute the square roots: \( \sqrt{16} = 4 \) and \( \sqrt{t^4} = t^2 \) because \((t^2)^2 = t^4\).

Step 4: Combine the results

Combine the results from Step 3 to get the final simplified expression as \( 4t^2 \).

Key Concepts

Square Roots

Square roots can appear mysterious, but they simply ask: "What number, when multiplied by itself, gives the original number?" For instance, the square root of 16 is 4, because 4 times 4 equals 16. This idea extends beyond numbers to algebraic terms too.
When dealing with expressions like \( \sqrt{16t^4} \), we break it down into smaller parts:\( \sqrt{16} \) and \( \sqrt{t^4} \). Calculating these separately can simplify the process: \( \sqrt{16} = 4 \) and \( \sqrt{t^4} = t^2 \), since \((t^2)(t^2) = t^4\).

• For numerical values, simply find what number squared equals the value.
• For variables with exponents, divide the exponent by 2 when taking the square root.
• This approach simplifies expressions step-by-step, making complex calculations manageable.

Algebraic Expressions

An algebraic expression combines numbers, variables, and operations (like addition or multiplication). When tasked with simplifying, the goal is to make this expression as straightforward as possible.
Take the example of \( \frac{80s^2t^4}{5s^2} \). To simplify, start by dividing the coefficients: \( \frac{80}{5} = 16 \). Then handle the variables: \( \frac{s^2}{s^2} = 1 \), meaning they effectively cancel each other out. What remains is \( 16t^4 \).

• Identify common terms in the numerator and denominator for simplification.
• Break down the expression into manageable parts before simplifying.
• This method streamlines the expression, leading to easier calculations or evaluations in future steps.

Variable Simplification

Variable simplification focuses on reducing expressions involving variables to their simplest form. This usually involves dividing, canceling, or factoring out terms to make solving or further simplifying easier. For example, simplifying \( \frac{s^2}{s^2} \) results in 1, since any non-zero number divided by itself equals 1.
When dealing with exponents, like in \( t^4 \), consider how exponents work. Here, \( t^4 \) can be expressed as \((t^2)^2\), which simplifies to \( t^2 \) after taking the square root.

• Cancel out identical terms in numerators and denominators.
• Understand that dividing variables with the same base involves subtracting their exponents.
• This helps reduce complexity, paving the way for easier evaluations or further operations.