Exponents, Logarithms & Radicals

How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$ ?

Let $n=8^{2022}$ . Which of the following is equal to $\frac{n}{4}?$

What is the value of $$(2^2-2)-(3^2-3)+(4^2-4)?$$

What is the value of $\sqrt{\left(3-2\sqrt{3}\right)^2}+\sqrt{\left(3+2\sqrt{3}\right)^2}$ ?

What is the value of $$2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?$$

What is the value of $$\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?$$

What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \tfrac{1}{2}$ ?

For what value $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$ ?

What is the value of $(2^0-1+5^2+0)^{-1}\times5?$

What is the value of $2-(-2)^{-2}$ ?

What is $\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}$ ?

What is the value of $\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?$

Assume that $x$ is a positive real number. Which is equivalent to $\sqrt[3]{x\sqrt{x}}$ ?

In the expression $c\cdot a^b-d$ , the values of $a$ , $b$ , $c$ , and $d$ are $0$ , $1$ , $2$ , and $3$ , although not necessarily in that order. What is the maximum possible value of the result?

The ratio $\frac{2^{2001}\cdot3^{2003}}{6^{2002}}$ is:

The ratio $\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ is closest to which of the following numbers?

$2000(2000^{2000}) = x$ Find x.

Ted mistakenly wrote $2^m\cdot\sqrt{\frac{1}{4096}}$ as $2\cdot\sqrt[m]{\frac{1}{4096}}.$ What is the sum of all real numbers $m$ for which these two expressions have the same value?

For how many (not necessarily positive) integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer?

Suppose that $P = 2^m$ and $Q = 3^n$ . Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$ ?

The fraction $$\frac{\left(3^{2008}\right)^2-\left(3^{2006}\right)^2}{\left(3^{2007}\right)^2-\left(3^{2005}\right)^2}$$ simplifies to which of the following?

Real numbers $a$ and $b$ satisfy the equations $3^{a} = 81^{b + 2}$ and $125^{b} = 5^{a - 3}$ . What is $ab$ ?

Which of the following is equivalent to $\sqrt{\frac{x}{1-\frac{x-1}{x}}}$ when $x < 0$ ?

What non-zero real value for $x$ satisfies $(7x)^{14}=(14x)^7$ ?

How many positive integers $n$ satisfy the following condition: $$\left(130n\right)^{50} > n^{100} > 2^{200} \ \text{?}$$

Find the value of $x$ that satisfies the equation $25^{-2} = \frac{5^{48/x}}{5^{26/x} \cdot 25^{17/x}}.$

Simplify $\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}$ .

Given that $3^8\cdot5^2=a^b,$ where both $a$ and $b$ are positive integers, find the smallest possible value for $a+b$ .

According to the standard convention for exponentiation, $$2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.$$ If the order in which the exponentiations are performed is changed, how many other values are possible?

The number $25^{64}\cdot 64^{25}$ is the square of a positive integer $N$ . In decimal representation, the sum of the digits of $N$ is

Positive integers $x$ and $y$ satisfy the equation $\sqrt{x} + \sqrt{y} = \sqrt{1183}$ . What is the minimum possible value of $x+y$ ?

What is the greatest integer less than or equal to $$\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?$$

Which of the following is equal to $\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}$ ?

Suppose that $4^a = 5$ , $5^b = 6$ , $6^c = 7$ , and $7^d = 8$ . What is $a \cdot b\cdot c \cdot d$ ?

Let $N$ be the positive integer $7777\ldots777$ , a $313$ -digit number where each digit is a $7$ . Let $f(r)$ be the leading digit of the $r{ }$ th root of $N$ . What is $$f(2) + f(3) + f(4) + f(5)+ f(6)?$$

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$ . How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $$5^n<2^m<2^{m+2}<5^{n+1}?$$