Problem 59 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form is \( y^6 \).

Step 1: Identify the Radical

The given expression is \( \sqrt{y^{12}} \). We need to simplify this expression by finding the square root of \( y^{12} \).

Step 2: Apply the Square Root Property

Recall that the square root of \( y^n \) is \( y^{n/2} \). Therefore, using this property, we simplify \( \sqrt{y^{12}} \) to \( y^{12/2} \).

Step 3: Simplify the Exponent

Now, calculate \( 12/2 \), which equals 6. So, \( y^{12/2} \) simplifies to \( y^6 \).

Step 4: Conclusion of Simplification

Thus, the simplified form of the original radical \( \sqrt{y^{12}} \) is \( y^6 \).

Key Concepts

Square Root

Understanding the concept of a square root is essential for solving radical expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 times 3 equals 9.
When you're working with square roots and variables, like \( \sqrt{y^{12}} \), it's helpful to remember the square root property for exponents:

• The square root of \( y^n \) is \( y^{n/2} \).

This rule helps you simplify expressions quickly by adjusting the power of the variable instead of dealing with cumbersome calculations. Remembering that squaring and square roots are inverse operations is crucial for successfully simplifying radicals. Begin by applying this property to separate and reduce the powers of variables when faced with more intricate expressions.

Exponents

Exponents are a way to express repeated multiplication of the same number or variable. For example, \( y^{12} \) means \( y \) multiplied by itself 12 times. When dealing with radicals like \( \sqrt{y^{12}} \), simplifying it involves understanding how to manipulate these exponents.
One imperative rule for exponents in the context of square roots is the division of the power by 2, as in \( y^{12/2} \). This step is just applying the square root rule, where you halve the exponent to find the square root.

• \( y^{12} \) becomes \( y^6 \) after simplifying because \( 12/2 = 6 \).

Mastering the properties of exponents, like this division rule, can make complex algebraic expressions much easier to manage, especially in exams or homework assignments.

Algebraic Expressions

Algebraic expressions involve variables and constants combined with arithmetic operations. Simplification of these expressions is a fundamental skill in algebra, often involving radicals and exponents.
In the expression \( \sqrt{y^{12}} \), we see both a radical and an exponent. Simplifying such expressions means rewriting them in a form that is easier to understand or compute. The goal is to reduce the expression to its simplest terms, which might involve eliminating radicals or balancing equations.

• This often includes applying laws of exponents, like dividing the exponent by 2 for square roots.
• The result, \( y^6 \), is simpler and more straightforward for further mathematical computations.

Being able to perform these simplifications is vital in problem-solving. Simplifying helps in identifying patterns, predicting outcomes, and making problems less daunting by breaking them into more manageable pieces.