Raise Powers to Powers
The Power Rule for Exponents
Another word for an exponent is power. You have likely seen or heard an example such as [latex]3^5[/latex] can be described as [latex]3[/latex] raised to the [latex]5[/latex]th power. In this section, we will further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, what to do when two numbers or variables are multiplied and both are raised to an exponent, and what to do when numbers or variables that are divided are raised to a power. We will begin by raising powers to powers.
the power rule for exponents
For any positive number [latex]x[/latex] and integers [latex]a[/latex] and [latex]b[/latex]: [latex]\left(x^{a}\right)^{b}=x^{a\cdot{b}}[/latex].
Raise a Product to a Power
Raising a product to a power is a fundamental operation in algebra that demonstrates how exponents interact with multiplication. This operation is widely used across various mathematical disciplines, including geometry, where it might be used to calculate the volume of shapes, and in finance, where it can be used to calculate compounded interest over multiple periods.
The rule simplifies the process of working with powers of products. Instead of multiplying the base numbers repeatedly, we apply the exponent to each factor individually. This is based on the distributive property of exponents over multiplication.
a product raised to a power
For any nonzero numbers [latex]a[/latex] and [latex]b[/latex] and any integer [latex]x[/latex], [latex]\left(ab\right)^{x}=a^{x}\cdot{b^{x}}[/latex].
If the variable has an exponent with it, use the Power Rule: multiply the exponents.
Raise a Quotient to a Power
Raising a quotient to a power is another key concept in algebra that involves exponents. This operation is essential when dealing with division in the context of exponential expressions. The power of a quotient rule tells us that when you raise a quotient to an exponent, you raise both the numerator and the denominator to that exponent separately.
This rule is incredibly useful for simplifying complex algebraic expressions, solving equations, and understanding geometric growth or decay when dealing with fractions or ratios
a quotient raised to a power
For any number [latex]a[/latex], any non-zero number [latex]b[/latex], and any integer [latex]x[/latex], [latex]\displaystyle {\left(\frac{a}{b}\right)}^{x}=\frac{a^{x}}{b^{x}}[/latex].