Problem 82
Question
Simplify. $$ \sqrt{128 q^{3}} $$
Step-by-Step Solution
Verified Answer
The simplified form is \ \[ 8 q \sqrt{2q} \].
1Step 1 - Factor the Radicand
To simplify the square root, start by factoring the expression under the square root. We have \ \( 128 q^3 \). Since 128 is not a perfect square, factor it to its prime factors: \ \(128 = 2^7\). Therefore, we can rewrite the radicand as \ \( 128 q^3 = 2^7 q^3 \).
2Step 2 - Apply the Product Rule for Square Roots
Next, use the property of square roots that states \( \ \sqrt{a \times b} = \ \sqrt{a} \times \ \sqrt{b} \ \) to separate the factors: \ \[ \sqrt{2^7 q^3} = \sqrt{2^7} \times \sqrt{q^3} \] \
3Step 3 - Simplify Each Square Root
Now, simplify each square root separately. \ \[ \sqrt{2^7} \] \ can be rewritten as \ \[ 2^{3.5} \] or \ \[ 2^3 \times \sqrt{2} \]. For \( \ \sqrt{q^3} \ \), we can rewrite it as \ \[ q^{1.5} = q^1 \times \sqrt{q} \].
4Step 4 - Combine the Simplified Terms
Finally, combine all the simplified terms together: \ \[ \sqrt{128 q^3} = \sqrt{2^7} \times \sqrt{q^3} = 2^3 \times \sqrt{2} \times q^1 \times \sqrt{q} \] \ \[ = 8 q \sqrt{2q} \]. Hence, the simplified form of \( \sqrt{128 q^3} \) is \ \[ 8 q \sqrt{2q} \].
Key Concepts
Factoring ExpressionsProduct Rule for Square RootsSimplifying Radicals
Factoring Expressions
Factoring expressions is one of the fundamental steps in simplifying mathematical expressions. To start with, when you are given a complex expression, breaking it down into its prime factors can make things much simpler.
For example, in the exercise to simplify \( \sqrt{128q^3} \), the first step is to factor the radicand (the expression inside the square root).
Here, 128 is not a perfect square, so we factor it into its prime components, which gives us:
\( 128 = 2^7 \).
This means the original expression can be rewritten as \( 128q^3 = 2^7q^3 \).
By factoring expressions into smaller components, it sets up the stage for the next steps and makes the simplification process easier.
For example, in the exercise to simplify \( \sqrt{128q^3} \), the first step is to factor the radicand (the expression inside the square root).
Here, 128 is not a perfect square, so we factor it into its prime components, which gives us:
\( 128 = 2^7 \).
This means the original expression can be rewritten as \( 128q^3 = 2^7q^3 \).
By factoring expressions into smaller components, it sets up the stage for the next steps and makes the simplification process easier.
Product Rule for Square Roots
The product rule for square roots is an essential property that helps in breaking down complex square roots. This rule states that the square root of a product is equal to the product of the square roots of each factor. Mathematically, it is represented as:
\( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
In our exercise, after factoring the expression, we apply the product rule to separate the factors:
\( \sqrt{2^7q^3} = \sqrt{2^7} \times \sqrt{q^3} \).
By using this property, we can handle each part of the expression separately, which simplifies our calculations in the next steps. With practice, this rule becomes a powerful tool to simplify various types of radical expressions.
\( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
In our exercise, after factoring the expression, we apply the product rule to separate the factors:
\( \sqrt{2^7q^3} = \sqrt{2^7} \times \sqrt{q^3} \).
By using this property, we can handle each part of the expression separately, which simplifies our calculations in the next steps. With practice, this rule becomes a powerful tool to simplify various types of radical expressions.
Simplifying Radicals
Simplifying radicals involves reducing a square root to its simplest form. After factoring the radicand and applying the product rule, the next step is to simplify each square root individually.
For instance, we take \( \sqrt{2^7} \) and simplify it further. Since \( 2^7 \) can be written as \( 2^{3.5} \), this breaks down to \( 2^3 \times \sqrt{2} \). Similarly, \sqrt{q^3} \ simplifies to \ q^{1.5} = q^1 \times \sqrt{q} \.
Finally, we combine all the simplified terms together: \( \sqrt{2^7q^3} = 2^3 \times \sqrt{2} \times q^1 \times \sqrt{q} = 8q \sqrt{2q} \).
The key to mastering simplifying radicals is to consistently apply these steps, ensuring you break down the problem into manageable pieces and then combine them for the final simplified form.
For instance, we take \( \sqrt{2^7} \) and simplify it further. Since \( 2^7 \) can be written as \( 2^{3.5} \), this breaks down to \( 2^3 \times \sqrt{2} \). Similarly, \sqrt{q^3} \ simplifies to \ q^{1.5} = q^1 \times \sqrt{q} \.
Finally, we combine all the simplified terms together: \( \sqrt{2^7q^3} = 2^3 \times \sqrt{2} \times q^1 \times \sqrt{q} = 8q \sqrt{2q} \).
The key to mastering simplifying radicals is to consistently apply these steps, ensuring you break down the problem into manageable pieces and then combine them for the final simplified form.