It looks like you're working on understanding exponents, specifically focusing on the rules for dividing with like bases, including the dividing rule and how to handle negative exponents. Below, I’ll help you with the vocabulary terms, provide an explanation of the dividing rule, and work through the five problems.
Vocabulary Terms
-
Exponent: A number indicating how many times to multiply the base by itself. For example, \(3^2\) means \(3 \times 3\).
-
Base: The number that is raised to a power. In the expression \(a^n\), \(a\) is the base.
-
Like Bases: Numbers that have the same base in an expression. For example, \(x^3\) and \(x^5\) are like bases.
-
Dividing Rule for Exponents: When dividing with like bases, subtract the exponent of the denominator from the exponent of the numerator. For example, \(a^m / a^n = a^{m-n}\).
-
Negative Exponent: Indicates the reciprocal of the base. For example, \(a^{-n} = 1/a^n\).
-
Expanded Form: Writing numbers as a sum of their parts, or expressing exponents as repeated multiplication. For example, \(x^3\) can be written as \(x \cdot x \cdot x\).
-
Power of a Power: When raising an exponent to another exponent, multiply the exponents. For example, \((a^m)^n = a^{mn}\).
-
Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, \(a^0 = 1\), where \(a ≠ 0\).
-
Reciprocal: The multiplicative inverse of a number. For example, the reciprocal of \(a\) is \(1/a\).
-
Equivalent Expressions: Different mathematical expressions that have the same value. For example, \(x^2 + 2x + 1\) is equivalent to \((x + 1)^2\).
Dividing Rule: Dividing with Like Bases
When you divide two expressions that have the same base, you use the formula:
\[ \frac{a^m}{a^n} = a^{m-n} \]
This is how the rule works: by subtracting the exponent of the denominator (bottom) from the exponent of the numerator (top), you simplify the expression while keeping the base the same.
Why do we subtract exponents when we are dividing like bases?
We subtract exponents when dividing like bases because we are essentially cancelling out common factors. For example, in \(x^5 / x^2\), you have \(x \cdot x \cdot x \cdot x \cdot x\) in the numerator and \(x \cdot x\) in the denominator. When you cancel the \(x \cdot x\) from both the top and the bottom, you are left with \(x^{3}\), which is what you get if you subtract \(2\) from \(5\) (i.e., \(5 - 2 = 3\)).
5 Problems - Show All Work
1. \(x^8 / x^2\)
\[ \frac{x^8}{x^2} = x^{8-2} = x^6 \]
2. \(x^{16} / x^{12}\)
\[ \frac{x^{16}}{x^{12}} = x^{16-12} = x^4 \]
3. \(x^{25} / x^{14}\)
\[ \frac{x^{25}}{x^{14}} = x^{25-14} = x^{11} \]
4. \(8x^6 / x^2\)
Assuming in this context \(8\) is a constant:
\[ \frac{8x^6}{x^2} = 8 \cdot x^{6-2} = 8x^4 \]
5. \(x^2 / x^4\)
\[ \frac{x^2}{x^4} = x^{2-4} = x^{-2} = \frac{1}{x^2} \]
What do I need help with?
Feel free to specify any topics or problems that you find confusing or need more explanation on, and I'll be glad to help!
1 answer