Problem 55 - College Algebra

Question

Simplify each expression. $$\sqrt{125 n^{10}}$$

Step-by-Step Solution

Answer

The simplified form is \(5n^5\sqrt{5}\).

Step 1: Prime Factorization

Start by factoring the number within the square root. The number 125 can be factored into primes as follows:\[ 125 = 5 \times 5 \times 5 = 5^3 \]

Step 2: Express Inside the Square Root

Express the entirety of the radicand using the factorization obtained in Step 1, combined with the given expression:\[ \sqrt{125n^{10}} = \sqrt{5^3 n^{10}} \]

Step 3: Apply the Square Root Property

Apply the property of square roots that states \( \sqrt{a^2} = a \) (for nonnegative \(a\)) to simplify \( \sqrt{5^3 n^{10}} \). Notice that:\[ 5^3 = 5^2 \times 5 = 25 \times 5 \]The square root of \(5^2\) is \(5\), so:\[ \sqrt{5^2} = 5 \]Additionally, apply the property to \(n^{10}\):\[ \sqrt{n^{10}} = n^{5} \] Now simplify the expression:\[ \sqrt{5^3 n^{10}} = 5n^5 \times \sqrt{5} \]

Step 4: Combine the Simplifications

Combine all parts to express the simplified form of the original expression:\[ 5n^5 \sqrt{5} \]This is the final simplified version of the original square root expression.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into the product of its prime numbers. This is an essential step when simplifying radicals, as it allows us to simplify the expression within the square root more easily.

To find the prime factors of a number, keep dividing the number by the smallest prime number that can go into it evenly, until you are left with only prime numbers. For instance, with the number 125:

• Start by dividing 125 by 5, the smallest prime number it is divisible by: 125 ÷ 5 = 25
• Next, divide 25 by 5: 25 ÷ 5 = 5
• Finally, divide 5 by 5: 5 ÷ 5 = 1

Thus, the prime factorization of 125 is 5 × 5 × 5, or expressed in exponential form, it's 5³. This step sets us up to easily simplify square roots.

Square Root Property

The square root property is a useful tool when simplifying expressions that contain square roots. The basic property states that \( \sqrt{a^2} = a \) for any nonnegative number \( a \). This property is fundamental because it allows us to separate perfect squares from under the square root sign, simplifying the expression.

In our example, once we express the radicand as \( 5^3n^{10} \), we can see parts that are perfect squares. Recognizing that \( 5^2 \) is a perfect square, it can be taken out from under the square root:

• Since \( \sqrt{5^2} = 5 \), the expression becomes simplified by removing \( 5 \).
• The term \( n^{10} \) can also be expressed as \( (n^5)^2 \), which means \( \sqrt{n^{10}} = n^5 \).

Thus, recognizing and applying this property helps us efficiently break down complex square root expressions.

Exponent Rules

Exponent rules are guidelines that help simplify expressions with powers or exponents. Knowing these rules allows us to handle more complicated algebraic expressions effectively.

One of the crucial rules when working with radicals is the power of a product rule: \((a^m)^n = a^{m \cdot n}\). This rule allows us to combine powers, making the simplification of radicals easier.

• In our instance, we disentangle \( n^{10} = (n^5)^2 \), using the rule \((a^m)^2 = a^{2m}\).
• We then express this understanding as \( \sqrt{(n^5)^2} = n^5 \), removing it directly from under the square root.

By applying these exponent rules, we streamline the process of simplifying radical expressions and reduce mechanical computation, ultimately reaching a simplified expression.