Problem 45 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \frac{\sqrt{30 p^{5} q^{14}}}{\sqrt{5 q^{7}}} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: $$\frac{\sqrt{30 p^{5} q^{14}}}{\sqrt{5 q^{7}}}$$ Answer: The simplified expression is: $$\sqrt{6\cdot p^{5}\cdot q^{7}}$$

Step 1: Rewrite the expression inside the square roots as a product of square roots

We will rewrite the given expression in the form that allows us to cancel out terms more easily. The square root of a product can be written as the product of two separate square roots. $$ \frac{\sqrt{30 p^{5} q^{14}}}{\sqrt{5 q^{7}}} = \frac{\sqrt{30}\sqrt{p^{5}}\sqrt{q^{14}}}{\sqrt{5}\sqrt{q^{7}}} $$

Step 2: Simplify common terms

Now that the expression has been separated into individual square roots, we can simplify the common terms, starting with the constant terms and moving on to the variables. $$ \frac{\sqrt{30}\sqrt{p^{5}}\sqrt{q^{14}}}{\sqrt{5}\sqrt{q^{7}}} = \frac{\sqrt{\frac{30}{5}}\sqrt{p^{5}}\sqrt{\frac{q^{14}}{q^{7}}}}{\sqrt{1}} $$

Step 3: Simplify the fraction inside the square roots

Simplify the fractions inside the square root: $$ \frac{\sqrt{\frac{30}{5}}\sqrt{p^{5}}\sqrt{\frac{q^{14}}{\underline{\phantom{xx}}}},q^{7}}{\sqrt{1}} = \frac{\sqrt{6}\sqrt{p^{5}}\sqrt{q^{7}}}{1} $$

Step 4: Combine the square roots

Combine the square roots back into a single square root: $$ \frac{\sqrt{6}\sqrt{p^{5}}\sqrt{q^{7}}}{1} = \sqrt{6\cdot p^{5}\cdot q^{7}} $$

Step 5: Write the final answer

The simplified expression is: $$ \sqrt{6\cdot p^{5}\cdot q^{7}} $$

Key Concepts

Understanding Square Roots

Square roots are an essential concept in algebra and mathematics as a whole. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 times 5 equals 25. In mathematical notation, the square root of a number \( x \) is represented as \( \sqrt{x} \). When simplifying expressions involving square roots, especially products or fractions, you can separate the square root over each part of the expression. This means \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \), which is helpful for simplifying complex expressions. For example, in the expression \( \frac{\sqrt{30 p^{5} q^{14}}}{\sqrt{5 q^{7}}} \), separate the square roots to simplify each term individually: \( \sqrt{30} \), \( \sqrt{p^{5}} \), and \( \sqrt{q^{14}} \). By doing this, you can easily identify common factors or terms that may be canceled out, simplifying the entire expression.

Rational Expressions and Their Simplification

Rational expressions are fractions where the numerator and the denominator are polynomials. Similar to simplifying numerical fractions, rational expressions can be simplified by reducing common factors from both the numerator and the denominator. Given a complex expression like \( \frac{\sqrt{30 p^{5} q^{14}}}{\sqrt{5 q^{7}}} \), consider each term under the square root as part of a larger rational expression.

• Start by identifying and reducing any common numerical factors. For instance, the numbers 30 and 5 have a common factor of 5, which simplifies \( \frac{30}{5} = 6 \).
• Next, handle any variables, in this case, \( q^{14} \) and \( q^{7} \), which can be simplified by dividing the powers, resulting in \( q^{14-7} = q^{7} \).

Simplifying rational expressions helps make the equation more manageable and easier to work with when solving math problems.

Exploring Exponents in Expressions

Exponents denote how many times a number, known as the base, is multiplied by itself. They follow specific rules that make operations more straightforward. For an expression like \( p^{5} \), the exponent shows that \( p \) is to be multiplied by itself 5 times: \( p \times p \times p \times p \times p \).When you have exponents in square root expressions, you can simplify them using the rule that says \( \sqrt{x^{n}} = x^{n/2} \). This means the square root effectively halves the exponent. For example, \( \sqrt{p^{5}} \) can be simplified by expressing it as \( p^{5/2} \). Moreover, when multiplying or dividing like bases, you handle the exponents by adding or subtracting them. Hence, simplifying a term like \( q^{14} \) divided by \( q^{7} \) results in \( q^{14-7} = q^{7} \). Understanding and applying the rules of exponents allows for easier simplification of complicated algebraic expressions, making solutions efficient and elegant.