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Ever wondered how to write a giant number like a million without filling a page? That's where exponents come in! They're like tiny superheroes, saving us time and space. An exponent represents how many times a number, called the base, is multiplied by itself. For example,5^3 means 5 \times 5 \times 5. Here, 5 is the base and 3 is the exponent.
To add expressions with exponents, it’s important to remember that you can only add like terms (terms with the same base and exponent). For example:
Step 1: Ensure the bases and exponents match. You have 2^3 + 2^3.
Step 2: Add the coefficients (if not visible, the coefficient is 1). So, 1(2^3) + 1(2^3) = 2(2^3).
Step 3: Simplify the result if possible, which remains 2(2^3) = 2 \times 8 = 16.
You can't add 2² (4) and 3² (9) because they have different bases (2 and 3).
When multiplying exponents and the bases are the same, you add the exponents. For instance:
Step 1: Write down the multiplication, e.g., 3^2 \times 3^3.
Step 2: Add the exponents since the bases (3) are the same. 2 + 3 = 5.
Step 3: The result is 3^5, which is 243.
When dividing exponents with the same base, subtract the exponent in the denominator from the exponent in the numerator. Here’s how:
Step 1: Set up your expression, e.g., 5^6 \div 5^2.
Step 2: Subtract the lower exponent from the higher exponent. 6 - 2 = 4.
Step 3: Write the result as 5^4, which equals 625.
Simplifying exponents involves using rules of exponents to make expressions easier to manage. For example:
Step 1: Identify opportunities to apply exponent rules, such as (2^3)^2 .
Step 2: Multiply the exponents (3 and 2), as the rule (a^m)^n = a^{m \times n}.
Step 3: Simplify the expression to 2^6, which calculates to 64.
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