Problem 98 - Prealgebra

Question

Simplify. $$7 \cdot 8^{2}+2 \cdot 5^{2}$$

Step-by-Step Solution

Answer

The simplified expression is 498.

Step 1: Simplify Exponents

First, compute the exponents independently. The term \(8^2\) means we multiply 8 by itself: \(8 \times 8 = 64\). Similarly, \(5^2\) means we multiply 5 by itself: \(5 \times 5 = 25\). Now, the expression becomes \(7 \cdot 64 + 2 \cdot 25\).

Step 2: Multiply the Coefficients

Next, multiply the coefficients by the results from Step 1. Calculate \(7 \cdot 64 = 448\) and \(2 \cdot 25 = 50\). Substitute these products back into the expression, which updates it to \(448 + 50\).

Step 3: Perform the Final Addition

Finally, add the results from Step 2 together: \(448 + 50 = 498\). This is the simplified form of the original expression.

Key Concepts

Exponents

An exponent is a number that shows how many times we multiply the base number by itself. For example, in the expression \(8^2\), 8 is the base, and 2 is the exponent. This tells us we need to multiply 8 by itself once, resulting in \(8 \times 8 = 64\).
Similarly, for \(5^2\), the base is 5 and the exponent is 2, so we calculate \(5 \times 5 = 25\).
Understanding exponents is crucial because they simplify larger operations into more manageable ones. Instead of multiplying a number over and over, we use this shorthand to clearly communicate the repeated multiplication. Exponents are a core part of algebra and empower us to solve equations efficiently.

Multiplication of Numbers

Multiplication is a basic arithmetic operation where we combine equal groups. When operating with expressions like \(7 \cdot 8^2\) and \(2 \cdot 5^2\), it's key to tackle the multiplication after any exponents.
In the problem, we've computed the exponents first and then used multiplication to further simplify the expression. Starting with \(7 \cdot 64\), we multiply 7 groups of 64, while for \(2 \cdot 25\), it's 2 groups of 25.
This results in \(7 \times 64 = 448\) and \(2 \times 25 = 50\).
Breaking down multiplication with clear, step-by-step calculations makes handling larger numbers and complex expressions much easier. It also reinforces the relationship between multiplication and addition—showing how 7 groups of 64 is adding 64 to itself 7 times.

Order of Operations

Order of operations is a fundamental principle in mathematics dictating the sequence in which we perform operations. This order ensures consistency and accuracy across all computations.
The acronym PEMDAS is a useful mnemonic: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). In our exercise \(7 \cdot 8^2 + 2 \cdot 5^2\), we first handled the exponents (\(8^2\) and \(5^2\)), moved on to multiplication, and concluded with addition.

• Start with any exponents.
• Proceed with multiplication or division, as applicable.
• Finish with addition or subtraction.

This methodical approach can solve even the trickiest of expressions and is a critical tool for mastering algebra and beyond. Knowing the order of operations avoids common pitfalls and errors when simplifying mathematical expressions.