Problem 67
Question
Simplify. $$ \sqrt{p^{3}} $$
Step-by-Step Solution
Verified Answer
The simplified form is \(p \sqrt{p}\).
1Step 1: Break down the expression
The expression inside the radical is \(p^3\). This can be written as \(p^2 \times p\).
2Step 2: Apply property of square roots
Use the property \(\forall a, b > 0: \sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Thus, \(\sqrt{p^2 \times p} = \sqrt{p^2} \times \sqrt{p}\).
3Step 3: Simplify each part separately
\(\sqrt{p^2}\) simplifies to \(p\) because \(p^2\) is a perfect square. Thus, \(\sqrt{p^2} = p\).
4Step 4: Combine simplified expressions
Combine the simplified parts to get \(p \times \sqrt{p}\). This is the simplified form of \(\sqrt{p^3}\).
Key Concepts
properties of square rootsexponentsperfect squares
properties of square roots
Understanding the properties of square roots is essential when simplifying radicals. One important property is that the square root of a product can be split into the product of the square roots of each factor.
Let's use some notation to make it clearer:
In our example, we applied this property to the expression \(\fracp^2 \times p\) which breaks down into \(\fracp^2 \times p\).
Understanding and applying this property is fundamental when working with radicals.
Let's use some notation to make it clearer:
- For any non-negative numbers, a and b, \(\forall a, b \, > 0: \sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)
In our example, we applied this property to the expression \(\fracp^2 \times p\) which breaks down into \(\fracp^2 \times p\).
Understanding and applying this property is fundamental when working with radicals.
exponents
Exponents are another key concept when simplifying radicals. An exponent tells you how many times to multiply a number by itself. For instance, \(a^2 = a \times a\) and \(a^3 = a \times a \times a\).
When working with square roots, it is helpful to recognize how exponents reduce. The square root of a number squared returns the base number. For example, \( \sqrt{a^2} = a\).
In our problem, \(p^3\) can be written as \( p^2 \times p \). Breaking it down like this allows us to apply the properties of square roots:
When working with square roots, it is helpful to recognize how exponents reduce. The square root of a number squared returns the base number. For example, \( \sqrt{a^2} = a\).
In our problem, \(p^3\) can be written as \( p^2 \times p \). Breaking it down like this allows us to apply the properties of square roots:
- \texample: \( \sqrt{p^2 \times p} = \sqrt{p^2} \times \sqrt{p}\)
perfect squares
A perfect square is a number that can be expressed as the product of an integer multiplied by itself. For example, \( 1, 4, 9, 16, \) etc.
In our solution, \( p^2 \) was identified as a perfect square since it is formed by multiplying \( p \times p \.\). Because of this, the square root of \( p^2\) simplifies to \( p \), making the overall simplification straightforward.
- example: \( 4 = 2 \times 2 \)
- another example: \( 9 = 3 \times 3 \)
In our solution, \( p^2 \) was identified as a perfect square since it is formed by multiplying \( p \times p \.\). Because of this, the square root of \( p^2\) simplifies to \( p \), making the overall simplification straightforward.