Pattern Behaviors in Exponents - PSAT Math

The powers of i form a sequence that repeats every four terms.

i1 = i

i2 = -1

i3 = -i

i4 = 1

i5 = i

Thus:

i25 = i

i23 = -i

i21 = i

i19= -i

Now we can evalulate the expression.

i25 - i23 + i21 - i19 + i17..... + i

= i + (-1)(-i) + i + (-1)(i) ..... + i

= i + i + i + i + ..... + i

Each term reduces to +i. Since there are 13 terms in the expression, the final result is 13i.

Multiplying like bases means add the exponents, so x+4 = 12, or x = 8.

Raising a power to a power means multiply the exponents, so 3y = 15, or y = 5.

x - y = 8 - 5 = 3.

The first step is to express both sides of the equation with equal bases, in this case 3. The equation becomes 33p = 32q. So then 3p = 2q, and q = (3/2)p is our answer.

272/3 is 27 squared and cube-rooted. We want to pick the easier operation first. Here that is the cube root. To see that, try both operations.

272/3 = (272)1/3 = 7291/3 OR

272/3 = (271/3)2 = 32

Obviously 32 is much easier. Either 32 or 7291/3 will give us the correct answer of 9, but with 32 it is readily apparent.

To solve this problem, we will have to take the log of both sides to bring down our exponents. By doing this, we will get aast logleft ($\frac{1}{3}$ right )= bast logleft ( 27 right ).

To solve for $\frac{a}{b}$ we will have to divide both sides of our equation by log$\frac{1}{3}$ to get $\frac{a}{b}$=\frac{logleft ( 27 right )}{logleft ( frac{1}${3} right )}.

$\frac{logleft ( 27 right )}{logleft ( frac{1}${3} right )} will give you the answer of –3.

We use two properties of logarithms:

log(xy) = log (x) + log (y)

$log(x^{n}$) = nlog (x)

So

left ( $\frac{2}{3}$ right $)^{x+1}$ = $\frac{27}{8}$ = left ( $\frac{3}{2}$ right $)^{3}$ = left ( $\frac{2}{3}$ right $)^{-3}$

which means

Remember the laws of exponents. In particular, when the base is nonzero:

An effective way to compare these statements, is to convert them all into exponents with base 2. The original statement becomes:

This is identical to statement I. Now consider statement II:

Therefore, statement II is not identical to the original statement. Finally, consider statement III:

which is also identical to the original statement. As a result, only I and III are the same as the original statement.

Using properties of radicals e.g.,

we get

Properties of Radicals

Properties of Radicals

Properties of Radicals

Convert the given expression into a single radical e.g. the expression inside the radical is:

and the cube root of this is :

Hence must be equal to 2.

Now

Hence the correct answer is

If we combine into a single logarithmic function we get:

Solving for we get .

The powers of i form a sequence that repeats every four terms.

i1 = i

i2 = -1

i3 = -i

i4 = 1

i5 = i

Thus:

i25 = i

i23 = -i

i21 = i

i19= -i

Now we can evalulate the expression.

i25 - i23 + i21 - i19 + i17..... + i

= i + (-1)(-i) + i + (-1)(i) ..... + i

= i + i + i + i + ..... + i

Each term reduces to +i. Since there are 13 terms in the expression, the final result is 13i.

Multiplying like bases means add the exponents, so x+4 = 12, or x = 8.

Raising a power to a power means multiply the exponents, so 3y = 15, or y = 5.

x - y = 8 - 5 = 3.