Problem 89 - Elementary Algebra

Question

Simplify. $$ \sqrt{300 p^{9} q^{11}} $$

Step-by-Step Solution

Answer

The simplified expression is \[ 10 p^4 q^5 \sqrt{3pq} \]

Step 1 - Factor inside the square root

Break down the expression inside the square root into factors that are easy to simplify. For instance, for the constant 300, we can express it as the product of its prime factors: \[ 300 = 2^2 \times 3 \times 5^2 \]. Now, the given expression can be written as: \[ \ \sqrt{300 p^9 q^{11}} = \sqrt{2^2 \times 3 \times 5^2 \times p^9 \times q^{11}} \]

Step 2 - Apply the square root to individual factors

Next, apply the square root to each factor separately: \[ \sqrt{2^2} \times \sqrt{3} \times \sqrt{5^2} \times \sqrt{p^9} \times \sqrt{q^{11}} \]. This simplifies to: \[ 2 \times \sqrt{3} \times 5 \times \sqrt{p^9} \times \sqrt{q^{11}} = 10 \sqrt{3} \times \sqrt{p^9} \times \sqrt{q^{11}} \]

Step 3 - Simplify the variables inside the square root

Next, take square roots of the variable parts. For this, write the exponents as sums of even and one less even number plus the remaining part if necessary.\[ \sqrt{p^9} = \sqrt{p^8 \times p} = \sqrt{(p^4)^2 \times p} = p^4 \sqrt{p} \], and\[ \sqrt{q^{11}} = \sqrt{q^{10} \times q} = \sqrt{(q^5)^2 \times q} = q^5 \sqrt{q} \]

Step 4 - Combine simplified parts

Combine all simplified parts to form the final expression: \[ 10 \times p^4 \times q^5 \times \sqrt{3} \times \sqrt{p} \times \sqrt{q} = 10 p^4 q^5 \sqrt{3pq} \]

Key Concepts

Prime Factorization

Prime factorization breaks a number down into its simplest parts, called prime numbers. Prime numbers are numbers greater than 1 and only divisible by 1 and themselves.
For example, to factorize 300, we find that 300 can be expressed as:

• Start with the smallest prime, 2, and divide: 300 ÷ 2 = 150
• Divide 150 by 2: 150 ÷ 2 = 75
• The next smallest prime is 3: 75 ÷ 3 = 25
• Finally, break 25 into 5s: 25 = 5 × 5

Putting this all together, we get: \[ 300 = 2^2 \times 3 \times 5^2 \] This helps in simplifying square roots as each of these prime factors can be separately managed.

Square Root Properties

Understanding square root properties can make simplifying expressions easier. A square root essentially 'undoes' squaring. For instance: \[ \sqrt{a^2} = a \]

• This property allows us to individually apply the square root to each factor within a product.

Example: \[ \sqrt{300 \times p^9 \times q^{11}} = \sqrt{2^2 \times 3 \times 5^2 \times p^9 \times q^{11}} \]
By breaking it down, we can simplify each part:

• \[ \sqrt{2^2} = 2 \]
• \[ \sqrt{5^2} = 5 \]
• \[ \sqrt{p^9} \]
• \[ \sqrt{q^{11}} \]

Leading to further simplification.

Simplifying Exponents

Exponents simplify repeated multiplication. To manage square roots with exponents, break the exponents into sums of even numbers and others. Consider: \[ \sqrt{p^9} \]
To simplify, rewrite it as: \[ \sqrt{p^9} = \sqrt{p^8 \times p} = \sqrt{(p^4)^2 \times p} \]
The square root then becomes: \[ \sqrt{(p^4)^2} \times \sqrt{p} = p^4 \sqrt{p} \]
You can apply similar steps for: \[ \sqrt{q^{11}} = \sqrt{(q^5)^2 \times q} = q^5 \sqrt{q} \] Using these steps, you simplify complex powers inside square roots.

Algebra Steps

Putting everything together involves a few algebraic steps:

• Break down the expression into prime factors
• Apply the square root to each factor separately
• Simplify the expressions involving exponents
• Combine all parts together for the final result

Example:
For the expression \[ \sqrt{300 p^9 q^{11}} = \sqrt{2^2 \times 3 \times 5^2 \times p^9 \times q^{11}} \]
Apply the square root step-by-step:

• \[ 10 \sqrt{3} \]
• \[ p^4 \sqrt{p} \]
• \[ q^5 \sqrt{q} \]

Combining it: \[ 10 p^4 q^5 \sqrt{3pq} \]