Problem 116 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{108 q^{10}}{3 q^{2}}} $$

Step-by-Step Solution

Answer

6 q^{4}

Step 1: Simplify the fraction under the square root

First, simplify the fraction inside the square root: \[ \frac{108 q^{10}}{3 q^{2}} \] Divide both the numerator (108) and the denominator (3) by their greatest common factor, which is 3: \[ \frac{108 ÷ 3 ~ q^{10}}{3 ÷ 3 ~ q^{2}} = \frac{36 q^{10}}{q^{2}} \]

Step 2: Simplify the variables

Next, simplify the variable part of the fraction: \[ \frac{36 q^{10}}{q^{2}} \] Subtract the exponents (since the base is the same, both are q): \[ 36 q^{10 - 2} = 36 q^{8} \]

Step 3: Apply the square root

Now, apply the square root to the simplified expression: \[ \text{Take the square root of } 36 q^{8}: \] The square root of 36 is 6, and the square root of \( q^{8} \) is \( q^{4} \): \[ \text{Therefore, } \sqrt{36 q^{8}} = 6 q^{4} \]

Key Concepts

Fraction Simplification

Simplifying fractions is a very important skill in algebra.
It helps to make expressions easier to work with.
To simplify a fraction, divide both the numerator (the top part) and the denominator (the bottom part) by their greatest common factor (GCF).
The GCF is the largest number that divides both the numerator and the denominator without leaving a remainder.

For example, in the exercise:

• We have the fraction \[ \frac{108 q^{10}}{3 q^{2}} \] inside the square root.
• The GCF of 108 and 3 is 3.
• So, divide both by 3 to get \[\frac{108 \div 3 \ q^{10}}{3 \div 3 \ q^{2}} = \frac{36 q^{10}}{q^{2}} \]

This simplifies the fraction, making the next steps easier.

Exponent Rules

Exponent rules are very useful when simplifying expressions with variables.
Knowing these rules helps in performing operations such as multiplication and division with exponents.
Let's look at some key rules:

• Product of Powers: \[ a^{m} \times a^{n} = a^{m+n} \]
• Quotient of Powers: \[ \frac{a^{m}}{a^{n}} = a^{m-n} \]
• Power of a Power: \[ (a^{m})^{n} = a^{m \times n} \]
• Zero Exponent: \[ a^{0} = 1 \]

For instance, in the exercise:

• We simplify \[ \frac{36 q^{10}}{q^{2}} \] by using the quotient of powers rule: \[ {q^{10-2}} = q^{8} \]
• This gives us \[ 36 q^{8} \]

Square Roots

A square root is a number that, when multiplied by itself, gives the original number.
The square root symbol is \[ \sqrt{} \].
Some important properties of square roots include:

• The square root of a product: \[ \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \]
• The square root of a quotient: \[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \]

In our exercise, we need to find the square root of \[ 36 q^{8} \. \]

• The square root of 36 is 6, because \[ 6 \cdot 6 = 36 \]
• The square root of \[ q^{8} \] is \[ q^{4} \] because \[ (q^{4}) \cdot (q^{4}) = q^{8} \]

Putting these together, we get \[ \sqrt{36 q^{8}} = 6 q^{4} \. \]