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Exponents are a way to represent repeated multiplication of the same number by itself. For example, when we see something like \(4^2\), it means \(4\) multiplied by itself, so \(4 \times 4\). It's important to be aware of where exponents are placed in an expression and how they affect the calculation. Here's a simple breakdown:

• The base is the number being multiplied (4 in this case).
• The exponent (or power) tells you how many times to multiply the base by itself (2 in this case).

In expressions with exponents, you need to take special care of how operations are structured, especially if there are parentheses involved.If we take the expression \(5 \cdot 4^2\), the exponent applies only to the 4, meaning we calculate \(4 \times 4 = 16\) first, and then multiply the result by 5, resulting in 80.Remember, always resolve exponents before multiplication, unless parentheses dictate otherwise.

Parentheses play a crucial role in mathematical expressions by indicating which operations should be performed first. They can completely change the order in which you perform calculations.Consider the expression \((5 \cdot 4)^2\). Here, the parentheses tell us to first solve what's inside them before considering any exponent outside. Let's see how this works:

• First, perform the operation inside the parentheses: \(5 \cdot 4 = 20\).
• Then, do the operation outside, which is the exponent, so \(20^2 = 20 \times 20 = 400\).

By using parentheses, you're enclosing different parts of the expression to be evaluated first, which gives precedence over other operations like exponents or multiplication that are outside the parentheses.This demonstrates how essential parentheses are in controlling the order of operations, ensuring clarity in solving mathematical problems.

In the order of operations, multiplication plays a key role and is evaluated after operations like parentheses and exponents have been resolved. It's important to clearly understand when and in what order multiplication occurs within different expressions.Taking the expression \(5 \cdot 4^2\), the calculation is:

• First, resolve the exponent: \(4^2\), which equals 16.
• Then, multiply the result by 5: \(5 \times 16 = 80\).

For the expression \((5 \cdot 4)^2\), the multiplication occurs first inside the parentheses:

• Multiply 5 by 4 inside the parentheses: \(5 \times 4 = 20\).
• Then, take the result and apply the exponent: \(20^2 = 400\).

Always remember that while multiplication is a straightforward operation, its placement within the order of operations can affect the entire outcome of an expression. Hence, be sure to follow preceding operations dictated by exponents and parentheses before multiplying.