COPY: Algebra Properties of Exponents and Rational Exponents

Explain how the power of a power property can be used to simplify the expression ((x^2)^3).

The expression simplifies to (x^{2 \cdot 3} = x^6) using the power of a power property.

What is the simplified form of (\frac{a^5}{a^{2}}) using the quotient of powers property?

Using the quotient of powers property, the expression simplifies to (a^{5-2} = a^3).

Convert the rational exponent expression (27^{\frac{2}{3}}) into radical form.

The expression can be written as (\sqrt[3]{27^2}) or (\left(\sqrt[3]{27}\right)^2).

If (\sqrt{x+8} = 4), what is the value of (x) after solving the equation?

Squaring both sides yields (x+8 = 16), so (x = 8). Signup and view all the answers

How can you simplify the expression (x^{\frac{2}{5}} \cdot x^{\frac{3}{5}})?

By using the product of powers property, it simplifies to (x^{\frac{2}{5} + \frac{3}{5}} = x^{1} = x). Signup and view all the answers

What is the result of applying the negative exponent property to the expression $a^{-3}$?

$\frac{1}{a^3}$ Signup and view all the answers

How would you simplify the expression $\frac{b^4}{b^2}$ using the quotient of powers property?

$b^2$ Signup and view all the answers

How can you express the rational exponent $16^{\frac{1}{4}}$ in radical form?

$\sqrt[4]{16}$ Signup and view all the answers

What is the simplified expression for $(xy)^{3}$ using the power of a product property?

$x^3y^3$ Signup and view all the answers

If $x^{\frac{5}{2}} = 27$, what is the value of $x$?

$9$ Signup and view all the answers

How would you simplify $\left(\frac{2}{3}\right)^{-2}$?

$\frac{9}{4}$ Signup and view all the answers

What value does $a^0$ take for any non-zero value of $a$?

$1$ Signup and view all the answers

Describe the steps involved in solving the radical equation $\sqrt{x-1} = 3$.

Isolate the radical, square both sides to get $x-1 = 9$, then solve for $x$: $x = 10$. Signup and view all the answers

What is the result of $\left(a^{2}\right)^{4}$ using the power of a power property?

$a^8$ Signup and view all the answers

How do you simplify the expression $x^{\frac{3}{4}} \cdot x^{\frac{1}{4}}$?

$x^1$ Signup and view all the answers