Problem 82 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \sqrt{16 x^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 4x^2 \).

Step 1: Identify the Expression

The expression given is \( \sqrt{16x^4} \). Our goal is to simplify this expression and express it using rational exponents.

Step 2: Express the Square Root Using Exponents

Recall that a square root can be expressed using exponents as \( x^{1/2} \). Thus the expression \( \sqrt{16x^4} \) can be rewritten as \( (16x^4)^{1/2} \).

Step 3: Apply the Power Rule

The power rule for exponents states that \( (a^m)^n = a^{m\cdot n} \). Apply this rule to each term in the expression: \[ (16)^{1/2} \text{ and } (x^4)^{1/2}. \]

Step 4: Simplify the Coefficient

Simplify \( (16)^{1/2} \). Since 16 is a perfect square, \( (16)^{1/2} = 4 \).

Step 5: Simplify the Variable

For the variable part, use the power rule: \( (x^4)^{1/2} = x^{4 \cdot 1/2} = x^2 \).

Step 6: Combine the Simplified Parts

Combine the simplified coefficient and variable parts to produce the final simplified expression, which is \( 4x^2 \). Therefore: \[ \sqrt{16x^4} = 4x^2 \]

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They form the basis of most algebra problems and are very versatile. In our example, the expression given was \( \sqrt{16x^4} \), a combination of a constant (16) and a variable part \( x^4 \) inside a square root.

When working with algebraic expressions:

• Identify each component. Here, 16 is a constant, and \( x^4 \) is the algebraic term.
• Understand operations and relationships. The square root impacts both the constant and the algebraic term.
• Manipulate them using algebraic rules to simplify or rearrange them.

Algebraic expressions allow us to express complex relationships simply and to perform calculations that help solve various problems in math.

Square Roots

The square root is a specific kind of root where the goal is to find a number which, when multiplied by itself, gives the original number. When none of the numbers involved are negative, the square root operation is entirely straightforward.

For instance, the square root of 16 is 4, because 4 times 4 equals 16. This is used in our solution where we simplified \( \sqrt{16} \) to get 4. When dealing with variables like \( x^4 \), the rule remainsthe same: find the value that can be multiplied by itself to return \( x^4 \).

Square roots can also be expressed using exponents. Instead of writing \( \sqrt{x} \), we can write it as \( x^{1/2} \). This is a crucial step in simplifying expressions like \( \sqrt{16x^4} \) using rational exponents.

Exponent Rules

Exponent rules are the guidelines that help us simplify and manipulate expressions with powers. These rules make working with complex algebraic expressions much simpler.

One important rule is the power rule, which states \((a^m)^n = a^{m\cdot n}\). This means when you have an exponent raised to another exponent, you multiply the powers.

In our solution, we applied the power rule to simplify \((x^4)^{1/2}\) into \(x^{2}\). Understanding and applying these rules can help simplify or transform expressions to solve problems.

• The product rule (\(a^m \times a^n = a^{m+n}\)) combines terms with the same base.
• The quotient rule (\(a^m / a^n = a^{m-n}\)) divides terms with the same base.
• The power rule simplifies terms raised to another power.
• These rules also allow us to handle negative and fractional exponents.

Knowing and applying these exponent rules enables students to manipulate algebraic expressions more easily, simplifying their calculations and understanding of math problems.