Lesson 6.2: Operations, Exponents, and Roots

Introduction

In this lesson, we will explore the essential operations, exponents, and roots that form the basis of arithmetic on the GRE. Understanding these concepts will help you tackle various quantitative reasoning problems efficiently. We will cover order of operations, work with fractions and decimals, and understand the rules governing exponents and radicals. By the end of this lesson, you will have the skills to simplify expressions, execute multi-step arithmetic with confidence, and apply exponent and radical rules correctly.

Learning Objectives

• Understand and apply the order of operations.
• Conduct arithmetic operations with fractions and decimals.
• Master the rules for exponents and radicals.
• Simplify expressions involving powers and roots.
• Execute multi-step arithmetic accurately and efficiently.
• Apply exponent and radical rules correctly.

Order of Operations

In mathematics, the order of operations is a set of rules that dictates the sequence in which different operations should be performed to ensure that everyone arrives at the same result. The mnemonic PEMDAS is often used to remember this order:

1. Parentheses
2. Exponents
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Example 1: Order of Operations

Consider the expression:

$$ 4 + 2 \times (3^2 - 1) $$

Step 1: Evaluate inside the parentheses first.

Calculate $3^2$, which is $9$, then subtract $1$:

$$ 3^2 - 1 = 9 - 1 = 8 $$

Step 2: Replace the parentheses in the original expression:

$$ 4 + 2 \times 8 $$

Step 3: Perform multiplication next:

$$ 2 \times 8 = 16 $$

Step 4: Finally, perform addition:

$$ 4 + 16 = 20 $$

Thus, the result of the expression is $20$.

Arithmetic with Fractions

Fractions represent a part of a whole and consist of a numerator and a denominator. To perform arithmetic with fractions, you need to understand how to add, subtract, multiply, and divide them.

Adding and Subtracting Fractions

When adding or subtracting fractions, the denominators must be the same (they must have a common denominator).

Example 2: Adding Fractions

Calculate:

$$ \frac{1}{4} + \frac{1}{2} $$

Step 1: Find a common denominator. The least common denominator (LCD) of $4$ and $2$ is $4$.

Step 2: Rewrite $\frac{1}{2}$ as $\frac{2}{4}$.

Thus, the expression becomes:

$$ \frac{1}{4} + \frac{2}{4} $$

Step 3: Add the numerators:

$$ \frac{1 + 2}{4} = \frac{3}{4} $$

Therefore, $\frac{1}{4} + \frac{1}{2} = \frac{3}{4}$.

Multiplying and Dividing Fractions

When multiplying fractions, simply multiply the numerators and the denominators. When dividing, multiply by the reciprocal of the second fraction.

Example 3: Multiplying Fractions

Calculate:

$$ \frac{2}{3} \times \frac{4}{5} $$

Multiply the numerators:

$$ 2 \times 4 = 8 $$

Multiply the denominators:

$$ 3 \times 5 = 15 $$

Thus,

$$ \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} $$

Example 4: Dividing Fractions

Calculate:

$$ \frac{3}{4} \div \frac{2}{5} $$

This is the same as multiplying by the reciprocal:

$$ \frac{3}{4} \times \frac{5}{2} $$

Multiply the numerators:

$$ 3 \times 5 = 15 $$

Multiply the denominators:

$$ 4 \times 2 = 8 $$

Thus,

$$ \frac{3}{4} \div \frac{2}{5} = \frac{15}{8} $$

Arithmetic with Decimals

Decimals are another way to represent fractions. The principle of arithmetic with decimals is similar to that of whole numbers, but you must keep track of the decimal point.

Adding and Subtracting Decimals

When adding or subtracting decimals, align the decimal points before performing the operation.

Example 5: Adding Decimals

Calculate:

$$ 3.75 + 0.6 $$

Align the decimal point:

$$egin{array}{r}

3.75 \

$+0.60 \ \hline$

$\end{array}$ $$

Then add:

$$ 3.75 + 0.60 = 4.35 $$

Multiplying and Dividing Decimals

When multiplying, treat the decimals as integers and then place the decimal point in the answer based on the total number of decimal places in both numbers.

Example 6: Multiplying Decimals

Calculate:

$$ 2.5 \times 0.3 $$

Multiply as if they were whole numbers:

$$ 25 \times 3 = 75 $$

Now count the decimal places. There is one decimal place in $2.5$ and one in $0.3$, for a total of two:

$$ 0.75 $$

Example 7: Dividing Decimals

Calculate:

$$ 4.5 \div 1.5 $$

Convert to a whole number by multiplying both the numerator and denominator by $10$:

$$ \frac{45}{15} $$

Now divide:

$$ 45 \div 15 = 3 $$

Rules for Exponents

Exponents represent repeated multiplication of a number by itself. Understanding the rules for exponents is crucial for simplifying expressions.

The Basic Rules of Exponents

1. Multiplying with the same base:

$$ a^m \cdot a^n = a^{m+n} $$

1. Dividing with the same base:

$$ \frac{a^m}{a^n} = a^{m-n} $$

1. Power of a power:

$$ (a^m)^n = a^{mn} $$

1. Product of powers:

$$ (ab)^n = a^n \cdot b^n $$

1. Quotient of powers:

$$ \left(\frac{a}{b}

ight)^n = $\frac{a^n}{b^n}$ $$

1. Zero exponent:

$$ a^0 = 1 $$

(provided a

eq 0)

1. Negative exponent:

$$ a^{-n} = \frac{1}{a^n} $$

Example 8: Using Exponent Rules

Simplify:

$$ 2^3 \cdot 2^2 $$

Using the rule for multiplying with the same base:

$$ 2^3 \cdot 2^2 = 2^{3+2} = 2^5 = 32 $$

Rules for Radicals

Radicals, or roots, are the inverse operations of exponents. The most common radical is the square root, denoted as $\sqrt{x}$, which means the number that, when multiplied by itself, gives $x$.

Key Rules of Radicals

1. Multiplying Radicals:

$$ \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} $$

1. Dividing Radicals:

$$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $$

1. Power of a Radical:

$$ (\sqrt{a})^2 = a $$

Example 9: Simplifying Radicals

Simplify:

$$ \sqrt{50} $$

Step 1: Factor $50$ into its prime factors:

$$ 50 = 25 \cdot 2 = 5^2 \cdot 2 $$

Step 2: Use the multiplication rule:

$$ \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} $$

Step 3: Simplify:

$$ \sqrt{25} = 5 $$

Thus,

$$ \sqrt{50} = 5\sqrt{2} $$

Conclusion

Mastering operations, exponents, and roots is fundamental to succeeding in GRE quantitative reasoning. This lesson helped you develop a solid understanding of the order of operations, as well as the manipulation of fractions and decimals. We also covered exponent and radical rules that are essential for simplifying complex expressions. Understanding these topics will enable you to solve problems more efficiently and accurately.

Study Notes

• Remember PEMDAS for order of operations.
• Common denominator is essential for adding/subtracting fractions.
• Move carefully with decimals; align decimal points.
• Know all exponent and radical rules for simplification.
• Practice gives speed; focus on mental math to enhance estimation skills.