Problem 115 - 1,001 GED Practice Questions For Dummies

Question

Simplify \(5 \times\left(91^0+3^2\right)\). A) 35 B) 45 C) 50 D) 500

Step-by-Step Solution

Answer

C) 50

Step 1: Simplify the exponent

First, simplify the exponent \(91^0\). Using the rule that any non-zero number to the power of 0 is equal to 1, we can write \(91^0=1\) and rewrite the expression: \[5 \times\left(1+3^2\right).\]

Step 2: Simplify the expression inside the parentheses

Now we need to solve the expression inside the parentheses: \[1+3^2=1+9.\] Add the two numbers: \[1+9 = 10.\]

Step 3: Multiply the result by 5

Now we can multiply the result inside the parentheses by 5: \[5 \times 10 = 50.\] Thus, simplifying the expression \(5 \times\left(91^0+3^2\right)\) gives us the result: C) 50

Key Concepts

Exponent Rules

Understanding exponent rules is crucial for simplifying expressions that include powers. One of the fundamental rules is the zero exponent rule, which states that any non-zero base raised to the power of zero is equal to one. For example, in the expression \(91^0\), it doesn't matter how large the number 91 is, the result is always \(1\).

Another important exponent rule to remember is when multiplying powers with the same base, the exponents should be added together. For instance, \(x^a \times x^b = x^{a+b}\). Conversely, when dividing powers with the same base, you subtract the exponents. Remembering these simple rules can significantly simplify complex algebraic expressions. Moreover, when an exponent is raised to another exponent, like \( (x^a)^b \), you multiply the exponents to get \( x^{a \times b} \).

Practice applying these rules consistently to become proficient in dealing with powers in algebraic expressions.

Order of Operations

In mathematics, the order of operations is critical to correctly interpreting and simplifying expressions. It's commonly remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This standard ensures that everyone simplifies expressions the same way.

In the practice problem, \(5 \times\left(91^0+3^2\right)\), we first address the parentheses, which is why we simplify \(91^0\) before doing anything else. Once the contents of the parentheses are simplified, we proceed with multiplication.

Understanding and applying the correct order of operations will prevent errors and confusion when dealing with more involved algebraic expressions. It is especially important when working with expressions that have multiple steps and operations.

Basic Algebra

At the heart of algebra is the manipulation of equations and expressions through basic operations and rules. Simplifying expressions, as we did in the example with \(5 \times\left(91^0+3^2\right)\), is a fundamental skill in basic algebra. It involves combining like terms, using the distributive property, and applying exponent rules, all while following the order of operations.

In our problem, simplifying inside the parentheses is crucial, as we combine the values into a single term before multiplying. The expression simplification process always starts with resolving the most nested parts, often contained within parentheses, and then expands outward, applying arithmetic operations in the correct sequence.

Developing strong skills in basic algebra is essential for tackling more complex mathematical problems. Regular practice with a variety of expressions and adherence to algebraic principles will lead to a more intuitive understanding of algebra and problem-solving.