Problem 65 - Algebra A Combined Function

Question

Assume that all variables represent positive real numbers. $$ \sqrt{y^{12}} $$

Step-by-Step Solution

Answer

\( y^6 \)

Step 1: Identify the Power Under the Square Root

The expression is given as \( \sqrt{y^{12}} \). Notice that there is a power of 12 under the square root.

Step 2: Apply the Power of a Power Rule

Recall that \( \sqrt{x} \) is equivalent to \( x^{1/2} \). Therefore, \( \sqrt{y^{12}} = (y^{12})^{1/2} \).

Step 3: Simplify the Expression Using Exponents

Use the rule \((a^m)^n = a^{m \times n}\). Here, \((y^{12})^{1/2} = y^{12 \times (1/2)} = y^{6}\).

Step 4: Final Answer

The simplified expression of \( \sqrt{y^{12}} \) is \( y^6 \).

Key Concepts

Exponent Rules

Exponent rules are fundamental when it comes to simplifying expressions with variables raised to powers. Understanding these rules makes it easier to manipulate expressions and solve problems. One primary rule used in the problem above is the power of a power rule. This rule states that when you have an exponent raised to another exponent, you multiply the exponents together:

• If you have \((a^m)^n\), it simplifies to \(a^{m \times n}\).

In our exercise, this rule is applied by taking \((y^{12})^{1/2}\) and simplifying to \(y^{12 \times (1/2)}\), which results in \(y^{6}\). Remembering these rules helps to quickly reduce complex expressions in mathematics.

Square Roots

Square roots are another crucial concept when working with radical expressions. The square root of a number or an expression is a value that, when multiplied by itself, gives the original number or expression.

• The notation \(\sqrt{x}\) is equivalent to \(x^{1/2}\).

This indicates that for any expression inside the square root, you are effectively raising that expression to the power of \(1/2\). Considering this definition helps simplify expressions under a square root, especially when combined with exponent rules. In the example of \(\sqrt{y^{12}}\), recognizing \(\sqrt{y^{12}} = (y^{12})^{1/2}\) allows us to use exponent rules to further simplify it.

Positive Real Numbers

In mathematics, assuming all variables are positive real numbers can simplify calculations and interpretations significantly. Positive real numbers are all the numbers greater than zero and are not imaginary. This assumption can make algebraic manipulations, such as square roots, much easier since we avoid complications introduced by negative numbers or variables.

• For positive real numbers, the square root operation is straightforward and does not result in complex numbers.

In the problem we solved, recognizing that \(y\) is a positive real variable means that taking \(\sqrt{y^{12}}\) simplifies to a real number without additional constraints or considerations. The simplification to \(y^{6}\) becomes direct given that exponent rules neatly apply without the need for further clarifications.