Problem 64 - Algebra 2

Question

Simplify. \(\sqrt{144 z^{8}}\)

Step-by-Step Solution

Answer

The simplified form is \(12z^4\).

Step 1: Identify the Square Root

The expression given is \(\sqrt{144 z^8}\). We need to find the square root of both the number 144 and the variable expression \(z^8\).

Step 2: Simplify the Square Root of the Number

Find the square root of 144, which is a perfect square. This is \(\sqrt{144} = 12\).

Step 3: Simplify the Square Root of the Variable Expression

For the variable part \(z^8\), recall that \(\sqrt{z^n} = z^{n/2}\). Thus, \(\sqrt{z^8} = z^{8/2} = z^4\).

Step 4: Combine Simplified Parts

Combine both simplified parts to write the complete simplified expression: \(\sqrt{144 z^8} = 12z^4\).

Key Concepts

Simplifying Expressions

Simplifying expressions means rewriting them in a more compact or accessible form without changing their value. It helps to make complex expressions easier to work with or understand. For instance:

• A complicated expression like \(2x + 3x\) can be simplified to \(5x\) because you combine like terms.
• Similarly, the expression \(\frac{8}{16}\) simplifies to \(\frac{1}{2}\) because you reduce the fraction by dividing both numerator and denominator by their greatest common divisor.

In mathematical expressions, we often deal with numbers and variables. To simplify these expressions, it’s important to identify parts of the expression that can be combined or rewritten in a simpler form. This involves knowing basic operation rules, like adding and subtracting similar terms, dividing to simplify fractions, and applying exponent rules. Simplifying helps in solving equations and understanding relationships between variables, forming a crucial foundation in algebra and higher mathematics.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. Simplifying radical expressions can make them more manageable and clearer to work with in equations and other mathematical operations. When dealing with square roots as in our example, the primary focus is on finding a number which, when multiplied by itself, gives the original number.

• For instance, the square root of 16 is 4 because \(4 \times 4 = 16\).
• The cube root of 27 is 3, as \(3 \times 3 \times 3 = 27\).

It's important to recognize perfect squares because they significantly simplify the process. In our example, simplifying \(\sqrt{144 z^8}\) involves recognizing that 144 is a perfect square and \(z^8\) can be simplified using exponent rules. With practice, identifying these properties speeds up and clarifies the process of working with radical expressions.

Exponents

Exponents are a shorthand way to express repeated multiplication of a number by itself. For example, \(z^8\) means multiplying \(z\) by itself eight times. Understanding how to manipulate expressions with exponents is crucial for simplifying them effectively.

• The rule \(a^m \cdot a^n = a^{m+n}\) helps in multiplying expressions with the same base.
• Similarly, \(\frac{a^m}{a^n} = a^{m-n}\) aids in division.
• Most importantly in simplification, as seen in \(\sqrt{z^8}\), the rule \(\sqrt{z^n} = z^{n/2}\) comes into play.

This means, instead of trying to multiply out all eight \(z\)’s, we can instantly simplify \(\sqrt{z^8}\) to \(z^4\). Exponent rules like this save time and reduce complexity, especially in algebra and calculus problems. Mastery of exponents allows you to tackle a wide array of mathematical challenges, making them a cornerstone of mathematical proficiency.