exponent

10, 1,000, 100,000, 10,000,000 What power of 10 is the 80th term? We see the following pattern 10^1 = 10 10^3 = 1000 10^5 = 100,000 10^7 = 10,000,000 f(n) = 10^(2n - 1) We build the 80th term: f(80) = 10^(2(80) - 1) f(80) = 10^(160 - 1) f(80) = 10^159

1, 8, 27, 64 What is the 10th term? We see the following pattern: 1^3 = 1 2^3 = 8 3^3 = 27 4^3 = 64 We build our sequence function using this pattern: f(n) = n^3 With n = 10, we have: f(10) = 10^3 f(10) = 1,000

2^n = 4^(n - 3) 2^n = (2^2)^(n - 3) (2^2)^(n - 3) = 2^2(n - 3) 2^n= 2^2(n - 3) Comparing exponents, we see that: n = 2(n - 3) n = 2n - 6 Subtract n from each side: n - n = 2n - n - 6 0 = n - 6 n = 6

5^(n - 1) = 15,625 We know 5^6 = 15,625, so we have: n - 1 = 6 Add 1 to each side: n - 1 + 1 = 6 + 1 Cancel the 1's on the left side: n = 7

if x^2=y^3, for what value of z does x^{3z}= y^9 y^9 = y^3 * y^3, so if we square the right side, we must square the left side for equivalence: x^2 * x^2 = x^4 Therefore, x^4 = y^9 Going back to our problem, x^{3z}= y^9, so 3z = 4 Divide each side by 3 to isolate z, and we have: 3z/3 = 4/3...

raise b to the 6th power then find the sum of the result and 4 raise b to the 6th power b^6 raise b to the 6th power then find the sum of the result and 4 b^6 + 4

3x to the power 2n We take the expression 3x raise it to the power of 2n (3x)^2n

The square of the difference of n and 2, increased by twice n The difference of n and 2: n - 2 The square of the difference of n and 2 means we raise (n - 2) to the 2nd power: (n - 2)^2 Twice n means we multiply n by 2: 2n The square of the difference of n and 2, increased by twice n (n -...

c varies jointly as the square of q and cube of p The square of q means we raise q to the 2nd power: q^2 The cube of p means we raise p to the rdd power: p^3 The phrase varies jointly means there exists a constant k such that: c = kp^3q^2

The cube of x is less than 15 The cube of x means we raise x to the 3rd power: x^3 Less than 15 means we setup the following inequality x^3 < 15

How many times bigger is 3^9 than 3^3 Using exponent rules, we see that: 3^9 = 3^3 * 3^6 So our answer is 3^6 times bigger

the square of the sum of 2a and 3b the sum of 2a and 3b 2a + 3b The square of this sum means we raise 2a + 3b to the 2nd power: (2a + 3b)^2

-x squared We take -x and raise it to the 2nd power: (-x)^2 = -x * -x = x^2

the square of the sum of x and y is less than 20 The sum of x and y means we add y to x: x + y the square of the sum of x and y means we raise the term x + y to the 2nd power: (x + y)^2 The phrase is less than means an inequality, so we write this as follows: (x + y)^2 < 20

raise r to the 8th power then find the product of the result and 3 Raise r to the 8th power means we raise r with an exponent of 8: r^8 The product of the result and 3 means we muliply r^8 by 3 3r^8

r squared plus the product of 3 and s plus 5 r squared means we raise r to the power of 2 r^2 The product of 3 and s means we multiply s by 3: 3s plus 5 means we add 3s + 5 R squared plus means we add r^2: r^2 + 3s + 5

4(a)(a)(b)(b)(b) We have 2 a's and 3 b's. We write this as: 4a^2b^3

cube root of a number and 7 The phrase a number means an arbitrary variable, let's call it x: x Cube root of a number means we raise x to the 1/3 power: x^1/3 And 7 means we add 7: x^1/3 + 7

3 less than a number times itself The phrase a number means an arbitrary variable, let's call it x: x Itself means the same variable as above. So we have: x * x x^2 3 less than this means we subtract 3 from x^2: x^2 - 3

ratio of the squares of t and u Ratio is also known as quotient in algebraic expression problems. The square of t means we raise t to the power of 2: t^2 The square of u means we raise u to the power of 2: u^2 ratio of the squares of t and u means we divide t^2 by u^2: t^2/u^2