Algebra - SAT Math

The complex conjugate of a complex number is , so has as its complex conjugate. The sum of the two numbers is

The complex conjugate of a complex number is . Therefore, the complex conjugate of is ; add them by adding real parts and adding imaginary parts, as follows:

,

the correct response.

The complex conjugate of a complex number is . Therefore, the complex conjugate of is ; add them by adding real parts and adding imaginary parts, as follows:

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

Add this to the second term to obtain the desired third term:

.

A power of can be evaluated by dividing the exponent by 4 and noting the remainder. The power is determined according to the following table:

, so

, so

, so

, so

Substituting:

A power of can be evaluated by dividing the exponent by 4 and noting the remainder. The power is determined according to the following table:

, so

, so

, so

, so

Substituting:

Collect real and imaginary terms:

It can be easier to line real and imaginary parts vertically to keep things organized, but in essence, combine like terms (where 'like' here means real or imaginary):

Multiplying two exponents that have the same base is the equivalent of simply adding the exponents.

So is the same as , and if , then or .

Using exponents, 27 is equal to 33. So, the equation can be rewritten:

34_x_ + 6 = (33)2_x_

34_x_ + 6 = 36_x_

When both side of an equation have the same base, the exponents must be equal. Thus:

4_x_ + 6 = 6_x_

6 = 2_x_

x = 3

_a_4 = _a_2 * _a_2 and _b_6= _b_2 * _b_2 * _b_2

Therefore _a_4 + _b_6 = 35 * 35 + 52 * 52 * 52 = 1,225 + 140,608 = 141,833

Since we have two ’s in we will need to combine the two terms.

For this can be rewritten as

So we have .

Or

Divide this by :

Thus or

*Hint: If you are really unsure, you could have plugged in the numbers and found that the first choice worked in the equation.

Since the base is 5 for each term, we can say 2 + n =12. Solve the equation for n by subtracting 2 from both sides to get n = 10.

When you multiply exponents, you add the common bases:

y4 x7 + y5x6 + y18x4 + y3x26

The answer is 5.

8 + 2x+1 = 72

2x+1 = 64

2x+1 = 26

x + 1 = 6

x = 5

We can solve by converting all terms to a base of two. 4, 16, and 32 can all be expressed in terms of 2 to a standard exponent value.

We can rewrite the original equation in these terms.

Simplify exponents.

Finally, combine terms.

From this equation, we can see that .

We want to simplify 5_b_ – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) .

Notice that we can collect the –5(b–1) terms, because they are like terms. There are 5 of them, so that means we can write –5(b–1) – 5(b–1) – 5(b–1) – 5(b–1) – 5(b–1) as (–5(b–1))5.

To summarize thus far:

5_b_ – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–_1) – 5(_b–1) = 5_b +(–5(_b–_1))5

It's important to interpret –5(b–1) as (–1)5(b–1) because the –1 is not raised to the (b – 1) power along with the five. This means we can rewrite the expression as follows:

5_b_ +(–5(b–1))5 = 5_b_ + (–1)(5(b–1))(5) = 5_b_ – (5(b–1))(5)

Notice that 5(b–1) and 5 both have a base of 5. This means we can apply the property of exponents which states that, in general, abac = a b+c. We can rewrite 5 as 51 and then apply this rule.

5_b_ – (5(_b–1))(5) = 5_b – (5(_b–1))(51) = 5_b – 5(_b–_1+1)

Now, we will simplify the exponent b – 1 + 1 and write it as simply b.

5_b_ – 5(b–1+1) = 5_b – 5_b = 0

The answer is 0.

is equal to which is equal to . If we compare this to the original equation we get

Combining the powers, we get .

From here we can use logarithms, or simply guess and check to get .