Problem 37 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \frac{\sqrt{75 y^{10}}}{\sqrt{3 y^{4}}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{\sqrt{75y^{10}}}{\sqrt{3y^4}}\). Answer: \(5y^3\).

Step 1: Identify the square root of each number and variable.

We need to find the square root of 75 and 3, as well as the square roots of \(y^{10}\) and \(y^{4}\).

Step 2: Simplify the square root of the numbers.

Simplify \(\sqrt{75}\) and \(\sqrt{3}\). To do that, find the prime factorization of 75: 75 = 3 x 25 = 3 x 5 x 5. So, \(\sqrt{75} = 5\sqrt{3}\). The expression becomes, $$ \frac{5\sqrt{3} y^{5}}{\sqrt{3} y^{2}} $$ with \(y^{5}\) because \(\sqrt{y^{10}}= y^{\frac{10}{2}} =y^{5}\) and \(y^{2}\) because \(\sqrt{y^4} = y^{\frac{4}{2}}=y^{2}\).

Step 3: Simplify the expression.

$$\frac{5\sqrt{3}y^{5}}{\sqrt{3} y^{2}}$$ can be simplified by cancelling out the similar terms in the numerator and denominator. In this case, we can cancel out the \(\sqrt{3}\) and simplify the \(y\) variables as follows, $$ \frac{5\cancel{\sqrt{3}}y^{5}}{\cancel{\sqrt{3}}y^{2}}=\frac{5y^{5}}{y^{2}} $$

Step 4: Simplify the variable's exponents.

Now, simplify the exponents of the variable \(y\) using the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\). In this case, m=5 and n=2, $$ \frac{5y^{5}}{y^{2}} = 5y^{5-2} = 5y^3 $$ So, the simplified expression is: \(5y^3\).

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up a mathematical room; you want everything in its simplest form. To simplify an expression, you combine like terms and reduce complex parts into a more manageable single expression. For example, if you have a fraction or a square root, your job is to simplify these components.

When faced with a problem, always start by identifying similar parts like terms that contain similar bases or can be factored together. It often involves removing any parenthesis and making calculations that help you cancel out terms.

• Combine like terms wherever possible.
• Reduce fractions by cancelling out terms that appear in both the numerator and the denominator.
• Keep an eye out for common factors that can be simplified instantaneously.

Simplification makes mathematical expressions easier to handle and solve, which is often the eventual goal in algebra.

Square Roots

Square roots are a fundamental concept that transform multiplicative quantities into their base squared form. So when you hear 'square root', think of the number that, when multiplied by itself, yields your original number. In mathematical form, if you square root 9, you get 3, because 3 x 3 equals 9.

When dealing with expressions under a square root, you can simplify them before proceeding. Here's how:

• Look for perfect squares—numbers that are whole numbers when square rooted, like 4, 9, 16, etc.
• For variables under a square root such as \( y^{10} \), divide the exponent by 2 to streamline it. In this case, \( \sqrt{y^{10}} = y^{\frac{10}{2}} = y^5 \).
• Break down the numbers inside the square root using prime factorization, and extract squared terms out of the square root.

Mastering square roots helps you simplify expressions efficiently and gives you a solid foundation for further algebraic manipulation.

Exponent Rules

Exponent rules are shortcuts that allow us to manipulate expressions involving powers and roots. Understanding these rules makes tackling algebraic tasks like simplifying expressions much easier.

There are several basic rules for exponents that are essential:

• The product of powers rule: \( a^m \cdot a^n = a^{m+n} \)
• The quotient of powers rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• The power of a power rule: \( (a^m)^n = a^{m \cdot n} \)

Applying these rules to simplify expressions is crucial. In our example, using the quotient of powers rule allowed us to simplify \( \frac{y^5}{y^2} = y^{5-2} = y^3 \). The ability to recognize and aptly apply these rules can significantly streamline your work in algebra, helping turn complex expressions into simpler and more manageable forms.