Problem 1 - Elementary and Intermediate Algebra

Question

\(5 x^{2}\) is the______ root of \(25 x^{4}\) because \(\left(5 x^{2}\right)^{2}=25 x^{4}\). The _______ root of 216 is 6 because \(6^{3}=216\).

Step-by-Step Solution

Answer

The blanks are 'square' and 'cube' respectively.

Step 1: Identify the Problem

The exercise is asking us to determine the appropriate words that fit in the blanks based on the mathematical explanations provided. We need to understand the relationship between the two expressions in each case.

Step 2: Understand the First Blank

The first blank involves the expression \(5x^2\) being the root of \(25x^4\). The explanation provided is: \((5x^2)^2 = 25x^4\). This indicates that we are looking for the type of root where the square of \(5x^2\) results in \(25x^4\). This is known as the 'square root'.

Step 3: Understand the Second Blank

The second blank involves finding what root 6 is of 216. The explanation given is \(6^3 = 216\). This implies that when 6 is raised to the power of 3, we get 216, suggesting that 6 is the 'cube root' of 216.

Step 4: Fill in the Blanks

Based on the analysis:- For the first sentence, 'square root' fills the blank because \(5x^2\) is the square root of \(25x^4\).- For the second sentence, 'cube root' fills the blank because 6 is the cube root of 216.

Key Concepts

Understanding Square Roots

Square roots are fundamental concepts in algebra. To put it simply, finding the square root of a number involves identifying a value that, when multiplied by itself, gives the original number.
For example, consider the expression \(5x^2\) as the square root of \(25x^4\). Why is that? Because when you multiply \(5x^2\) by itself, you get \(25x^4\). This is represented mathematically as: \((5x^2)^2 = 25x^4\).
A few key points about square roots:

• The square root of a positive number is always positive or zero.
• Every positive number has two square roots: one positive (the principal square root) and one negative.
• For variables, square roots are applicable only if the expressions under the root are non-negative when dealing with real numbers.

These properties of square roots are essential in solving many algebraic problems and help us simplify expressions and equations.

Delving Into Cube Roots

Cube roots take things a step further by exploring what happens when a number is raised to the power of three. The cube root of a number is the value that, when multiplied by itself twice, gives the original number.
In the exercise example, 6 is identified as the cube root of 216, because: \(6^3 = 216\).
Understanding cube roots can be particularly helpful in various mathematical contexts. Here are some crucial points:

• Unlike square roots, cube roots of both positive and negative numbers are possible.
• The cube root of a positive number is positive; the cube root of a negative number is negative.
• Cube roots are often used in problems involving volumes and other three-dimensional measurements as they relate to real-world applications.

Grasping the concept of cube roots can greatly aid in solving polynomial equations and in understanding more complex algebraic expressions.

Exploring Exponents and Powers

Exponents and powers are integral to understanding roots in algebra. They signify how many times a number is multiplied by itself.
In our exercise, two main types of roots were presented: square roots and cube roots. These can be viewed as:

• Square roots involve exponents of 2, as seen in \( (5x^2)^2 = 25x^4 \).
• Cube roots involve exponents of 3, demonstrated by \((6)^3 = 216\).

The laws of exponents simplify multiplication and division of powers. For instance:

• The product of powers rule: \(a^m \times a^n = a^{m+n}\).
• The power of a power rule: \( (a^m)^n = a^{m \times n} \).
• The power of a product rule: \( (ab)^m = a^m \times b^m \).
• The zero exponent rule: any number (except zero) raised to the power of zero is 1 (\(a^0 = 1\)).

Understanding these rules makes working with mixed variable expressions more manageable. With these skills, students can efficiently handle a variety of algebraic expressions and equations.