- Work with expressions that include square roots and raising numbers to powers.
Manipulating Expressions Involving Square Roots and Powers
Manipulating expressions involving square roots and powers is a fundamental skill necessary for understanding more advanced calculus topics, such as integration and differentiation. Understanding how to manipulate square roots and powers allows you to simplify expressions and solve equations effectively. Some key ways to simplify these expressions are:
- The power rule for exponents: [latex](a^m)^n = a^{mn}[/latex]
- The product rule for exponents: [latex]a^m \cdot a^n = a^{m+n}[/latex]
- The quotient rule for exponents: [latex]\frac{a^m}{a^n} = a^{m-n}[/latex]
- Simplifying square roots:[latex]\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}[/latex]
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.
The Product Rule for Exponents
The Product Rule for Exponents is one of the essential rules in algebra that simplifies the process of working with powers. This rule is pivotal when dealing with exponential expressions, particularly when multiplying them. In essence, it tells us that when we multiply two exponents with the same base, we can simply add the exponents to get the new power of the base.
This rule is extremely useful in various mathematical and real-world applications, such as calculating compound interest, understanding scientific notation, or solving problems in physics and engineering. By using the Product Rule, we can manage and simplify complex expressions without the need for lengthy multiplication.
the product rule for exponents
For any number [latex]x[/latex] and any integers [latex]a[/latex] and [latex]b[/latex], [latex]\left(x^{a}\right)\left(x^{b}\right) = x^{a+b}[/latex].
To multiply exponential terms with the same base, add the exponents.
Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says “For any number [latex]x[/latex] and any integers [latex]a[/latex] and [latex]b[/latex].”
[latex](a^{3})(a^{7})[/latex]
When multiplying more complicated terms, multiply the coefficients and then multiply the variables.
[latex]5a^{4}\cdot7a^{6}[/latex]
The Quotient (Division) Rule for Exponents
The Quotient Rule for Exponents is as crucial as the Product Rule and serves as its counterpart for division. This rule assists in simplifying expressions where we have exponential terms with the same base being divided. It states that when you divide exponents with the same base, you can subtract the exponents.
This rule has significant practical applications, especially in fields that involve calculations of rates of change, decay, or growth when they are decreasing, such as in the case of depreciation in finance or radioactive decay in physics.
the quotient (division) rule for exponents
For any non-zero number [latex]x[/latex] and any integers [latex]a[/latex] and [latex]b[/latex]:
To divide exponential terms with the same base, subtract the exponents.
When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.
Simplifying Square Roots and Expressing Them in Lowest Terms
To simplify a square root means that we rewrite the square root as a rational number times the square root of a number that has no perfect square factors. The act of changing a square root into such a form is simplifying the square root.
The number inside the square root symbol is referred to as the radicand. So in the expression [latex]\sqrt{a}[/latex] the number [latex]a[/latex] is referred to as the radicand.
Before discussing how to simplify a square root, we need to introduce a rule about square roots.
the product rule for square roots
The square root of a product of numbers equals the product of the square roots of those number.
Given that [latex]a[/latex] and [latex]b[/latex] are nonnegative real numbers,
Using this formula, we can factor an integer inside a square root into a perfect square times another integer. Then the square root can be applied to the perfect square, leaving an integer times the square root of another integer. If the number remaining under the square root has no perfect square factors, then we’ve simplified the square root into its lowest terms.
A perfect square is an integer that can be expressed as the square of another integer. For example, [latex]16[/latex], [latex]25[/latex], and [latex]36[/latex] are perfect squares because they are [latex]4^2[/latex], [latex]5^2[/latex], and [latex]6^2[/latex], respectively.
How to: To simplify a square root the lowest terms when [latex]n[/latex] is an integer
- Step 1: Determine the largest perfect square factor of [latex]n[/latex], which we denote [latex]a^2[/latex].
- Step 2: Factor [latex]n[/latex] into [latex]a^2×b[/latex].
- Step 3: Apply [latex]\sqrt{a^2 \times b} =\sqrt{a^2} \times \sqrt{b}[/latex].
- Step 4: Write [latex]\sqrt{n}[/latex] in its simplified form, [latex]a\sqrt{b}[/latex].
Simplify [latex]\sqrt{180}[/latex] and express in lowest terms.
Simplify [latex]\sqrt{330}[/latex] and express in lowest terms.