Problem 1 - Intermediate Algebra

Question

Fill in the blanks. \(5 x^{2}\) 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

5x^2 is the square root of 25x^4, and 6 is the cube root of 216.

Step 1: Understanding the mathematical operation

We are given that \((5x^2)^2 = 25x^4\). This statement suggests that 5x^2, when squared, equals 25x^4. Thus, we need to determine what we call a number that, when squared, results in another number.

Step 2: Identifying the relationship for square roots

In mathematics, when a number \(a\) is such that \(a^2 = b\), \(a\) is called the square root of \(b\). Therefore, \(5x^2\) is the square root of \(25x^4\) because \((5x^2)^2 = 25x^4\).

Step 3: Understanding the cube root operation

We need to find what we call a number that, when raised to the power of 3, equals another number.

Step 4: Identifying the relationship for cube roots

Similarly, when a number \(c\) is such that \(c^3 = d\), \(c\) is called the cube root of \(d\). Therefore, 6 is the cube root of 216 because \(6^3 = 216\).

Step 5: Fill in the blanks

Based on our understanding, the blanks should be filled with "square" and "cube," respectively. Thus, 5x^2 is the square root of 25x^4 and 6 is the cube root of 216.

Key Concepts

Cube Root

A cube root of a number is a value that, when raised to the power of three, results in the original number. Think of it as the reverse process of cubing a number. For example, if you cube 6, that means you calculate:

• \(6 \times 6 \times 6 = 216\)

In this case, we say 6 is the cube root of 216, denoted as \( \sqrt[3]{216} = 6 \). Understanding cube roots is crucial in solving equations involving cubes, as it helps find the original base number before it was cubed.
To find a cube root, you are essentially asking: what number multiplied by itself three times gives me the number I'm looking for? Just like square roots, cube roots can be found for both perfect cubes and non-perfect cubes, but the latter may result in irrational numbers.

Exponents

Exponents are a shorthand way of expressing repeated multiplication. When we see a number like \(x^n\), it's telling us to multiply \(x\) by itself \(n\) times. For instance, when you have \(5x^2\), it means:

• \(5 \times x \times x\)

This operation amplifies the number and is useful in algebra and everyday calculations where large numbers or repetitive multiplications appear.
Exponents have certain rules that make calculations easier, such as the product of powers rule, which states that when multiplying like bases, you add the exponents:

• \(x^a \times x^b = x^{a+b}\)

Understanding these rules can simplify complex equations and make solving algebraic expressions less daunting. Moreover, mastering the basics of exponents lays the foundation for more advanced studies such as calculus.

Algebraic Expressions

Algebraic expressions are fundamental in mathematics as they involve numbers, variables, and operators. They can look daunting at first but are simply a way to represent relationships and calculations mathematically. For example, take the algebraic expression \(5x^2\).
Here, the number 5 is a constant, \(x\) is a variable that can assume different values, and the exponent 2 indicates repeated multiplication of \(x\). This versatility allows algebraic expressions to describe a wide range of real-world problems like speeds, areas, and quantities.
A key skill in working with algebraic expressions is the ability to simplify or manipulate them using operations such as addition, subtraction, multiplication, and division. Arithmetic with variables involves the same processes as arithmetic with numbers, but requires careful attention to maintaining equality and correctly applying laws such as the distributive property and combining like terms.