Question

Evaluate the expressions without using a calculator. $$ 5^{2} \cdot 2^{2} $$

Short answer

Answer: The value of the expression $5^{2} \cdot 2^{2}$ is 100.

Step-by-step solution

Calculate the values of the exponents separately.

First, we need to calculate the value of each exponent: $$ 5^{2} = 5 \cdot 5 = 25 $$ and $$ 2^{2} = 2 \cdot 2 = 4 $$

Multiply the results of the exponents.

Now that we have the values of each exponent, we can multiply them together to get our final answer: $$ 25 \cdot 4 = 100 $$

Write the answer.

Therefore, the value of the expression is: $$ 5^{2} \cdot 2^{2} = 100 $$

Key concepts

Multiplication

When evaluating expressions with multiplication, it’s important to understand how multiplication works. Multiplication is one of the basic arithmetic operations where you combine equal groups. If you have 5 sets of 4 apples, you can multiply 5 by 4 to find out how many apples you have in total. This is essentially what happens with numbers—except instead of apples, we are multiplying numerical values. An easy way to conceptualize multiplication is through repeated addition. For example, multiplying 5 by 2 means adding 5 to itself, or 5 + 5, which equals 10. In our exercise, once the exponents were calculated: \(5^2 = 25\) and \(2^2 = 4\), we multiply these two results: \(25 \times 4 = 100\). This gives us the total value of the initial expression.

Expressions

An expression in math is a combination of numbers, variables, and operators (like +, \, -, \, *, \, and /). Expressions allow us to formulate problems mathematically. They can be as simple as \(3 + 2\), or more complex, like \(5^2 \cdot 2^2\) as in your exercise.Understanding how to manipulate expressions is crucial as it forms the building blocks of more advanced mathematics. With expressions, you often follow steps to simplify or solve them. In this case, we concentrated on managing exponents and multiplication. Each part of \(5^2 \cdot 2^2\) was worked through separately before combining them to find the answer. Learning to break these expressions into manageable parts can significantly ease the overall solving process.

Arithmetic Operations

Arithmetic operations are fundamental to math and include addition, subtraction, multiplication, and division. These operations are used to perform calculations and solve problems. In our example, multiplication and exponentiation were the main operations. Understanding these operations means knowing their rules and how they interact. For example, exponentiation is a form of repeated multiplication: \(a^n = a \cdot a \cdot \, \ldots \, \cdot a\), \(n\) times. Calculating \(5^2\) involves multiplying 5 by itself, giving us 25. Similarly, \(2^2\) means 2 times itself, which is 4. After resolving each exponent separately, you use multiplication, another basic operation, to combine them: \(25 \cdot 4 = 100\).These operations are the foundation of arithmetic, making them essential skills on your math journey. Leveraging these can help perform a wide array of calculations whether they involve simple arithmetic or more complex expressions.