Simplifying Expressions - SAT Math

First, we can use the property of exponents that xy/xz = xy–z

Then we can use the property of exponents that states x–y = 1/xy

a–1b5c–1 = b5/ac

Cross multiply to get 14ay = 15bx, then divide by 15b to get x by itself.

Let f = Francine's age now.

e = f – 5

f = e + 5

In 2 years, Francine will be f + 2. Use our previous calculation to substitute.

f + 2 = (e + 5) + 2 = e + 7

Three consecutive positive integers are added together. If the largest of the three numbers is m, find the sum of the three numbers in terms of m.

If m is the largest of three consecutive positive integers, then the integers must be:

m – 2, m – 1, and m, where m > 2.

The sum of these three numbers is:

m - 2 + m – 1 + m = 3_m_ – 3

The difference between the price to repair the car or buy a new one is $11,000. You divide this by the monthly savings, $220, giving you 50 months.

In both expressions, the trinomial in can be seen to be a perfect square trinomial:

Taking the square root of both sides of this latter equation:

and

and

Taking the square root of both sides of this latter equation:

and

By transitivity,

Square both sides:

so one of two things is possible:

or

This makes

the correct choice.

Remember that dividing two fractions is the same as multiplying the numerator by the reciprocal of the denominator, like so:

Substitute 4 for x + y in the expression given.

4 minus 6 equals –2.

Write an equation for the written expression: 9x – 6 = 48. When we solve for x we get x = 6.

If we substitute 6 in for x in the given equation and set our answer to 2, we can solve for y algebraically. 30 minus 4y divided by y equals 2 **-->**2y =30 -4y --> 6y =30 --> y=5. We could also work from the answers and substitute each answer in and solve.

To simplify this expression, you must combine like terms. You should first use the distributive property and multiply -4 by x2 and -4 by 3.

x3 - 4x2 -12 + 15

You can then add -12 and 15, which equals 3.

You now have x3 - 4x2 + 3 and are finished. Just a reminder that x3 and 4x2 are not like terms as the x’s have different exponents.

Multiply each term of the first factor by each term of the second factor and then combine like terms.

(2x + 4)(x2 – 2x + 4) = 2x3 – 4x2 + 8x + 4x2 – 8x + 16 = 2x3 + 16

5x – (6x – 2x) = 5x – (4x) = x

(x – 1)(x + 2) - x2 + 2 = x2 + x – 2 – x2 + 2 = x

x(4x)/(4x) = x

(3 – 3)x = 0x = 0

Using d = rt, we know that first part of the trip can be represented by f = rg. The second part of the trip can be represented by 30 = rx, where x is some unknown number of hours. Note that the rate r is in both equations because Sophie is traveling at the same rate as mentioned in the problem.

Solve each equation for the time (g in equation 1, x in equation 2).

g = f/r

x = 30/r

The total time is the sum of these two times

Note that, from equation 1, r = f/g, so

=

Add the two equations:

a + b = 10

b + c = 15

------------

a + b + b + c = 10 + 15

a + 2_b_ + c = 25 (this is the denominator of the answer)

Subtract the two equations:

b + c = 15

a + b = 10

------------

b + c – (a + b) = 15 – 10

c – a = 5 (this is the numerator of the answer)

5/25 = 1/5

Eq 1: a = 2b

Eq 2: 3b = c

Eq 3: 2c = 3d

Rewrite Eq. 3 substituting using Eq. 2.

2(3b) = 3d (because c = 3b)

6b = 3d (simplify)

2b = d (divide by 3)

Since a and d both equal 2b, a = d. Therefore, d/a = 1.

If we were to multiply ab, bc, and ac together, we would get (ab)(bc)(ac) = a2b2c2 = (abc)2. If we were then to take the square-root of (abc)2, we would get abc, which is what the question asks us to find.

We know that ab = 4, bc = 5, and ac = 20. Thus (ab)(bc)(ac) = (4)(5)(20) = 400.

(ab)(bc)(ac) = a2b2c2 = 400.

(abc)2 = 400

abc = +√400 or –√400

abc = 20 or –20.

However, we are told that a, b, and c are all negative numbers, so the product of all three must be negative. Therefore, abc must be -20.

The answer is –20.

• 2√8 = 2√4 * √2 = 2 * 2√2 = 4√2
• 5√8 = 5√4 * √2 = 5 * 2√2 = 10√2
• 4√16 = 4 * 4 = 16

4√2 + 10√2 = 14√2

14√2 – 16

Begin by distributing each part:

2x(x2 + 4ax – 3a2) = 2x * x2 + 2x * 4ax – 2x * 3a2 = 2x3 + 8ax2 – 6a2x

The second:

–4a2(4x + 3a) = –16a2x – 12a3

Now, combine these:

2x3 + 8ax2 – 6a2x – 16a2x – 12a3

The only common terms are those with a2x; therefore, this reduces to

2x3 + 8ax2 – 22a2x – 12a3

This is the same as the given answer:

–12a3 – 22a2x + 8ax2 + 2x3