Fractional Exponents Revisited: A Complete Guide for Common Core Algebra 2 Students

# Fractional Exponents Revisited: A Complete Guide for Common Core Algebra 2 Students

If you are taking Common Core Algebra 2, you may have encountered fractional exponents in your homework. Fractional exponents are a way of writing roots and powers using a single notation. They can help you simplify expressions and solve equations involving radicals. In this article, we will review the basics of fractional exponents and show you some examples of how to use them.

## What are Fractional Exponents?

A fractional exponent is an exponent that is a fraction, such as $$\frac12$$ or $$\frac34$$. A fractional exponent can be written as a radical, or a root, using the following rule:

$$a^\fracmn=\sqrt[n]a^m$$

This means that to evaluate a fractional exponent, you first raise the base to the numerator power, and then take the denominator root of the result. For example:

$$4^\frac12=\sqrt[2]4^1=\sqrt4=2$$

$$8^\frac23=\sqrt[3]8^2=\sqrt[3]64=4$$

$$16^\frac34=\sqrt[4]16^3=\sqrt[4]4096=8$$

You can also write a radical as a fractional exponent using the same rule:

$$\sqrt[5]x^3=x^\frac35$$

$$\sqrt[2]y=\sqrty=y^\frac12$$

$$\sqrt[3]z^4=z^\frac43$$

## How to Simplify Expressions with Fractional Exponents?

One of the benefits of using fractional exponents is that they allow you to apply the same rules of exponents that you learned for integer exponents. For example, you can use the following rules to simplify expressions with fractional exponents:

- Product rule: $$a^m \cdot a^n = a^m+n$$

- Quotient rule: $$\fraca^ma^n = a^m-n$$

- Power rule: $$(a^m)^n = a^mn$$

- Negative rule: $$a^-m = \frac1a^m$$

Here are some examples of how to use these rules:

- Simplify $$\left(\fracx^\frac12x^\frac14\right)^\frac13$$

Using the quotient rule, we get:

$$\left(\fracx^\frac12x^\frac14\right)^\frac13=x^\frac12-\frac14)^\frac13=x^\frac14)^\frac13$$

Using the power rule, we get:

$$x^\left(\frac14\right)\left(\frac13\right)=x^\frac112$$

- Simplify $$\left(9x^-\frac32y^\frac56\right)^-\frac23$$

Using the power rule, we get:

$$\left(9x^-\frac32y^\frac56\right)^-\frac23=9^-\frac23x^-\left(-\frac32\right)\left(-\frac23\right)y^-\left(\frac56\right)\left(-\frac23\right)=9^-\frac23x^{\frac{-6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6+6-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-12-36

fractional exponents revisited common core algebra 2 homework

0$$, this means that $$x=1$$ is a minimum point. Therefore, -0.1875 is the lowest value of $$f(x)$$.

- Find the intercepts. To find the x-intercept, we set $$f(x)=0$$ and solve for $$x$$. We get two solutions: $$x=0$$ and $$x=8$$. These are the points where the graph crosses the x-axis. To find the y-intercept, we set $$x=0$$ and solve for $$f(x)$$. We get one solution: $$f(0)=-0.1875$$. This is

## How to Solve Equations with Fractional Exponents?

Another skill that you need to master in Common Core Algebra 2 is solving equations that involve fractional exponents. To solve these equations, we can use the following steps:

- Isolate the variable that has a fractional exponent. This means that we want to get rid of any other terms or factors that are not part of the fractional exponent expression. We can do this by using inverse operations, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

- Convert from a fractional exponent to a radical. This means that we want to rewrite the fractional exponent expression as a radical expression using the rule $$a^\fracmn=\sqrt[n]a^m$$. This will make it easier to apply the next step.

- Solve for the variable by using roots and/or exponents. This means that we want to undo the radical or the power by using the principle of powers. The principle of powers states that if two expressions are equal and have the same base, then their exponents must also be equal. For example, if $$a^m=a^n$$, then $$m=n$$. We can use this principle to raise both sides of the equation to a power that will eliminate the radical or the fractional exponent. For example, if we have $$\sqrtx=2$$, we can square both sides to get $$x=4$$. If we have $$x^\frac13=3$$, we can cube both sides to get $$x=27$$.

Here are some examples of how to solve equations with fractional exponents:

- Solve $$\frac14x^\frac43-\frac12x^\frac13=0$$

- Isolate the variable that has a fractional exponent. In this case, we have two terms that have fractional exponents with the same base and denominator. We can factor out the common factor of $$x^\frac13$$ to get $$x^\frac13\left(\frac14x-\frac12\right)=0$$

- Convert from a fractional exponent to a radical. We can rewrite $$x^\frac13$$ as $$\sqrt[3]x$$ using the rule $$a^\fracmn=\sqrt[n]a^m$$. We get $$\sqrt[3]x\left(\frac14x-\frac12\right)=0$$

- Solve for the variable by using roots and/or exponents. We can use the zero product property to set each factor equal to zero and solve for $$x$$. We get $$\sqrt[3]x=0$$ or $$\frac14x-\frac12=0$$. To solve for $$x$$ in the first equation, we can cube both sides to get $$x=0$$. To solve for $$x$$ in the second equation, we can add $$\frac12$$ to both sides and multiply by 4 to get $$x=2$$. Therefore, the solutions are $$x=0$$ or $$x=2$$.

- Solve $$(8/27)^-2/3=y^-2/3$$

- Isolate the variable that has a fractional exponent. In this case, we already have only one term on each side of the equation that has a fractional exponent with the same base and denominator. We do not need to do anything in this step.

- Convert from a fractional exponent to a radical. We can rewrite both sides of the equation as radical expressions using the rule $$a^\fracmn=\sqrt[n]a^m$$. We get $$\sqrt[3](8/27)^-2=\sqrt[3]y^-2$$

- Solve for the variable by using roots and/or exponents. We can use the principle of powers to raise both sides of the equation to a power that will eliminate the radicals and negative exponents. In this case, we can cube both sides and take the reciprocal of both sides to get $$(8/27)^-2=y^-2$$ and then $$(27/8)^2=y^2$$. To solve for $$y$$, we can take the square root of both sides and include both positive and negative solutions. We get $$y=\pm \sqrt(27/8)^2=\pm \frac278$$. Therefore, the solutions are $$y=\frac278$$ or $$y=-\frac278$$.

## How to Simplify Fractions with Fractional Exponents?

Another topic that you may encounter in Common Core Algebra 2 is simplifying fractions that have fractional exponents in the numerator or the denominator. To simplify these fractions, we can use the following steps:

- Rewrite any radicals as fractional exponents using the rule $$\sqrt[n]a^m=a^\fracmn$$

- Apply the properties of exponents to simplify any expressions with the same base in the numerator or the denominator. For example, use the product rule $$a^m \cdot a^n = a^m+n$$, the quotient rule $$\fraca^ma^n = a^m-n$$, and the power rule $$(a^m)^n = a^mn$$

- Reduce any common factors in the numerator and the denominator. For example, if both the numerator and the denominator have a factor of $$x^\frac12$$, we can divide both by $$x^\frac12$$ to cancel them out.

- Convert any fractional exponents back to radicals if needed.

Here are some examples of how to simplify fractions with fractional exponents:

- Simplify $$\frac\sqrt[3]x^2\sqrtx$$

- Rewrite any radicals as fractional exponents. We get $$\fracx^\frac23x^\frac12$$

- Apply the properties of exponents to simplify any expressions with the same base. In this case, we can use the quotient rule to subtract the exponents of $$x$$. We get $$x^\frac23-\frac12=x^\frac16$$

- Reduce any common factors in the numerator and the denominator. In this case, there are no common factors to reduce.

- Convert any fractional exponents back to radicals if needed. In this case, we can rewrite $$x^\frac16$$ as $$\sqrt[6]x$$

- Therefore, the simplified fraction is $$\sqrt[6]x$$

- Simplify $$\frac(4y)^\frac32(16y)^\frac14$$