Exponents and Logarithms

Expressions associated with exponent and base are called—

In $a^n$, $a$ is called—

In $a^n$, $n$ is called—

Successive multiplication of $n$ $a$'s equals—

For base $a \in \mathbb{R}$ and power $n \in \mathbb{Q}$, $a^n$ is—

Which set is denoted by $\mathbb{N}$?

Which set is denoted by $\mathbb{R}$?

Which set is denoted by $\mathbb{Q}$?

Irrational exponents are—

$2 \times 2 \times 2$ is written as—

$a \times a \times a$ in exponential form is—

$b \times b \times b \times b \times b$ equals—

$a^m \times a^n = ?$

$\dfrac{a^m}{a^n} = ?$ when $m > n$, $a \neq 0$:

$\dfrac{a^m}{a^n} = ?$ when $n > m$, $a \neq 0$:

$(ab)^n = ?$

$\left(\dfrac{a}{b}\right)^n = ?$ ($b \neq 0$):

$(a^m)^n = ?$

$2^3 \times 2^4 = ?$

$\dfrac{5^7}{5^3} = ?$

$\dfrac{3^2}{3^5} = ?$

$(2 \cdot 3)^4 = ?$

$\left(\dfrac{2}{3}\right)^3 = ?$

$(x^3)^2 = ?$

$a^3 \cdot a^{-3} = ?$

$(a^p)^q \cdot (a^q)^r \cdot (a^r)^p$ on simplification equals $a^?$:

$\dfrac{a^{m+n}}{a^n} = ?$

$(a^m)^n \cdot (a^n)^p$ equals—

$\dfrac{(ab)^3}{a^2 b}$ equals—

$\left(\dfrac{x^2}{y^3}\right)^4 = ?$

$\left(\dfrac{a^2 b}{c^3}\right)^2 = ?$

$a^m \cdot a^n \cdot a^p = ?$

$(2x^2)^3 = ?$

$\dfrac{a^7 b^3}{a^4 b^5} = ?$

Substituting "5" base, $5^m \cdot 5^n = 5^{m+n}$ illustrates—

Power of a quotient rule states—

$(a^2 b^3)^2 = ?$

$\dfrac{a^{2m}}{a^m} = ?$

$(x^a)^b \cdot (x^b)^a = ?$

$\dfrac{a^{n+1}}{a^n} = ?$

$a^0 = ?$ ($a \neq 0$):

$a^{-n} = ?$ ($a \neq 0, n \in \mathbb{N}$):

$5^0 = ?$

$0^0$ is—

$2^{-3} = ?$

$\left(\dfrac{1}{3}\right)^{-2} = ?$

$\dfrac{1}{a^{-n}} = ?$

$\dfrac{a^m}{a^m} = ?$

$\left(\dfrac{2}{3}\right)^5 \times \left(\dfrac{2}{3}\right)^{-5} = ?$

$\dfrac{5^2}{5^3} = ?$

$7^{-1} = ?$

For all integer $m, n$, $\dfrac{a^m}{a^n} = ?$

$(-3)^{-2} = ?$

$10^{-3} = ?$

$a^{-m} \cdot a^m = ?$

$5^{1/2}$ is written as—

$5^{1/3}$ is written as—

The $n^{th}$ root of $a$ is—

$\sqrt[n]{a}$ in exponent form is—

$5^{1/2} \times 5^{1/2} = ?$

$\sqrt[3]{a} \times \sqrt[3]{a} \times \sqrt[3]{a} = ?$

$a^{m/n} = ?$

$\left(a^{1/n}\right)^n = ?$

$\sqrt[4]{16} = ?$

$\sqrt[3]{27} = ?$

$8^{2/3} = ?$

$32^{1/5} = ?$

$\sqrt[5]{32^4} = ?$

Under condition $a > 0$, $a \neq 1$, if $a^x = a^y$ then—

Under $a > 0$, $b > 0$, $x \neq 0$, if $a^x = b^x$ then—

$\sqrt[6]{x^4 y^2 z^{-2}}$ in exponent form is—

$(125)^{-1/3} = ?$

$\sqrt{\sqrt{a}} = ?$

$(a^{1/2})^4 = ?$

$4^{1.5} = ?$

$\dfrac{5^2}{5^3} = ?$ (Example 1)

$\left(\dfrac{2}{3}\right)^5 \times \left(\dfrac{2}{3}\right)^{-5} = ?$ (Example 1)

Simplify $\dfrac{5^4 \cdot 8 \cdot 16}{2^7 \cdot 125}$ (Example 2):

Simplify $\dfrac{3 \cdot 2^n - 4 \cdot 2^{n-2}}{2^n - 2^{n-1}}$ (Example 2):

$(a^p)^{q-r} \cdot (a^q)^{r-p} \cdot (a^r)^{p-q} = ?$ (Example 3):

$(12)^{-2} \times \sqrt[4]{6^4}$ simplifies to (Example 4):

Solve $4^{x+1} = 32$:

$\dfrac{7 \times 7^{-3}}{7^4 \times 7^{-4}} = ?$

$(2^2 \cdot 4^{-1})^{1/2} = ?$

$\sqrt[6]{x^6} \cdot \sqrt[3]{y^3}$ simplifies to ($x, y, z > 0$):

$(3m)^{m-1} \cdot (3^{m-1})^{m+1}$ exponent of $3$ in product equals—

$\dfrac{x^{p+q}}{x^q} \cdot \dfrac{x^{q+r}}{x^r} \cdot \dfrac{x^{r+p}}{x^p} = ?$

If $a^x = b$, $b^y = c$, $c^z = a$, then $xyz = ?$

Solve $4^x = 8$:

Solve $2^{2x+1} = 128$:

Solve $(\sqrt{3})^{x+1} = (\sqrt{3})^{2x-1}$:

Solve $2^x + 2^{1-x} = 3$: