Exponents - ACT Math

Card 0 of 1854

Question

Which real number satisfies $3^{x}*9=27^{2}$ ?

Answer

Simplify the base of 9 and 27 in order to have a common base.

(3x)(9)=272

= (3x)(32)=(33)2

=(3x+2)=36

Therefore:

x+2=6

x=4

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Question

Which of the following is a factor of $4x^{4}+ 36 x^{2}$ ?

Answer

The terms of $4x^{4}+ 36 x^{2}$ have $4x^{2}$ as their greatest common factor, so

$4x^{4}+ 36 x^{2} = 4x^{2} (x^{2}+9)$

$x^{2}+ 9$ is a prime polynomial.

Of the five choices, only $x^{2}+ 9$ is a factor.

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Question

Which of the following expressions is equal to the following expression?

$\sqrt{(27)(45)(125)}$

Answer

$\sqrt{(27)(45)(125)}$

First, break down the component parts of the square root:

$\sqrt{(3^{3})(5\times 3^{2})(5^{3})}$

Combine like terms in a way that will let you pull some of them out from underneath the square root symbol:

$\sqrt{(5^{4})(3^4)(3)}$

Pull out the terms with even exponents and simplify:

$(5^{2})(3^{2})\sqrt{3}=225\sqrt{3}$

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Question

Which of the following is equal to the following expression?

$\sqrt{(16)(8)+(32)(20)}$

Answer

$\sqrt{(16)(8)+(32)(20)}$

First, break down the components of the square root:

$\sqrt{(2^{4})(2^{3})+(2^{5})(2^{2})\times 5}$

Combine like terms. Remember, when multiplying exponents, add them together:

$\sqrt{(2^{7})+(2^{7})\times5}$

Factor out the common factor of :

$\sqrt{(2^{7})(1+5)}$

$\sqrt{(2^7)\times6}$

Factor the :

$\sqrt{(2^7)\times2\times3}$

Combine the factored with the :

$\sqrt{(2^{8})\times3}$

Now, you can pull $\sqrt{2^8}$ out from underneath the square root sign as :

$2^4\sqrt{3}$

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Question

Which of the following expression is equal to

$\sqrt{(45)(9)+(27)(36)}$

Answer

$\sqrt{(45)(9)+(27)(36)}$

When simplifying a square root, consider the factors of each of its component parts:

$\sqrt{(3^2)\times5\times(3^{2})+(3^{3})(2^2)(3^2)}$

Combine like terms:

$\sqrt{5(3^4)+(2^2)(3^5)}$

Remove the common factor, :

$\sqrt{(3^4)\times(5+3\times(2^2))}$

Pull the $\sqrt{3^4}$ outside of the equation as :

$(3^2)\sqrt{5+12}=9\sqrt{17}$

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Question

Simplify $\frac{8^{3} \times 3}{2^{6}\times 27}$

Answer

The easiest way to approach this problem is to break everything into exponents. $8^{3}$ is equal to $2^{9}$ and 27 is equal to $3^{3}$ . Therefore, the expression can be broken down into $\frac{2^{9} \times 3}{2^{6}\times 3^{3}}$ . When you cancel out all the terms, you get $\frac{2^{3}}{3^{2}}$ , which equals $\frac{8}{9}$ .

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Question

What is,

$\frac{12}{\sqrt{18}}$ ?

Answer

To find an equivalency we must rationalize the denominator.

To rationalize the denominator multiply the numerator and denominator by the denominator.

$\frac{12}{\sqrt{18}}\frac{\sqrt{18}}{\sqrt{18}}=$

$\frac{12\sqrt{18}}{18}=$

Factor out 6,

$\frac{2\sqrt{18}}{3}=$

Extract perfect square 9 from the square root of 18.

$\sqrt{18}=3\sqrt{2}$

$\frac{23\sqrt{2}}{3}=$

$2*\sqrt{2}$

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Question

The solution of $\sqrt{x-3}>2$ is the set of all real numbers such that:

Answer

Square both sides of the equation: $x-3=2^{2}$

Then Solve for x:

Therefore,

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Question

What is the product of and

Answer

Multiplying complex numbers is like multiplying binomials, you have to use foil. The only difference is, when you multiply the two terms that have in the them you can simplify the to negative 1. Foil is first, outside, inside, last

First

$7i\cdot6=42i$

Outside:

$7i\cdot2i=14i^2=-14$

Inside

$-3\cdot6=-18$

Last

$-3\cdot2i=-6i$

Add them all up and you get

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Question

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Distribute:

Answer

This equation can be solved very similarly to a binomial like . Distribution takes place into both the real and nonreal terms inside the complex number, where applicable.

$(2\cdot4) + (2\cdot 7i)$

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Question

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Distribute and solve:

Answer

This problem can be solved very similarly to a binomial like .

$((5\cdot3)+ (5 \cdot i)) + ((2 \cdot 6) + (2\cdot 2i))$

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Question

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Which of the following is equivalent to ?

Answer

When dealing with complex numbers, remember that $i = \sqrt{-1}$ .

If we square , we thus get $i^2 = (\sqrt{-1})^2 = -1$ .

Yet another exponent gives us $i^3 = i^2 \cdot i = -1 \cdot i = -i$ OR $-\sqrt{-1}$ .

But when we hit , we discover that $i^4 = i^2 \cdot i^2 = -1 \cdot -1 = 1$

Thus, we have a repeating pattern with powers of , with every 4 exponents repeating the pattern. This means any power of evenly divisible by 4 will equal 1, any power of divisible by 4 with a remainder of 1 will equal , and so on.

Thus, $i^{23}$

$\frac{23}{4} = 5 \frac{3}{4}$

Since the remainder is 3, we know that $i^{23} = i^3 = -i$ .

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Question

Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.

Simplify the following expression, leaving no complex numbers in the denominator.

$\frac{3+2i}{4-4i}$

Answer

Solving this problem requires eliminating the nonreal term of the denominator. Our best bet for this is to cancel the nonreal term out by using the conjugate of the denominator.

Remember that for all binomials , there exists a conjugate such that $(a+b)\cdot(a-b) = (a^2-b^2)$ .

This can also be applied to complex conjugates, which will eliminate the nonreal portion entirely (since )!

$\frac{3+2i}{4-4i}$

$\frac{(3+2i)(4+4i)}{(4-4i)(4+4i)}$ Multiply both terms by the denominator's conjugate.

$\frac{(3+2i)(4+4i)}{(16-16i^2)} = \frac{(3+2i)(4+4i)}{32}$ Simplify. Note .

FOIL the numerator.

Combine and simplify.

$\frac{4+20i}{32} = \frac{1+5i}{8}$ Simplify the fraction.

Thus, $\frac{3+2i}{4-4i} = \frac{1+5i}{8}$ .

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Question

Which of the following is equal to ?

Answer

Remember that since $i = \sqrt{-1}{}$ , you know that is . Therefore, is or . This makes our question very easy.

is the same as or

Thus, we know that is the same as or .

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Question

Simplify the following:

Answer

Begin by treating this just like any normal case of FOIL. Notice that this is really the form of a difference of squares. Therefore, the distribution is very simple. Thus:

Now, recall that $i=\sqrt{-1}$ . Therefore, is . Based on this, we can simplify further:

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Question

Simplify the following:

Answer

Begin this problem by doing a basic FOIL, treating just like any other variable. Thus, you know:

Recall that since $i=\sqrt{-1}$ , . Therefore, you can simplify further:

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Question

An account is compounded at a given rate of interest annually for years. What is this rate if the beginning balance for the account was $\$4500$ and its ending balance $\$10451.43$ ? Round to the nearest hundredth of a percent.

Answer

Recall that the equation for compounded interest (with annual compounding) is:

Where is the balance, is the rate of interest, and is the number of years.

Thus, for our data, we need to know:

Now, let's use for . This gives us:

Using a logarithm, this can be rewritten:

$log_b(\frac{10451.43}{4500})=20$

This can be rewritten:

$\frac{log(2.32254)}{log(b)}=20$

Now, you can solve for :

$\frac{1}{log(b)}=\frac{20}{log(2.32254)}$

or

$log(b)=\frac{log(2.32254)}{20}$

Now, finally you can rewrite this as:

$10^\frac{log(2.32254)}{20}=b$

Thus,

Now, round this to and recall that

Thus, and or

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Question

Michelle makes a bank deposit of $1,500 at 4.2% annual interest, compounded monthly. Approximately how much money will be in Michelle’s bank account in 3 years?

Answer

The formula to use for compounded interest is

$A=P\left(1+\frac{r}{n} \right )^{nt}$

where P is the principal (original) amount, r is the interest rate (expressed in decimal form), n is the number of times per year the interest compounds, and t is the total number of years the money is left in the bank. In this problem, P=1,500, r=0.042, n=12, and t=3.

By plugging in, we find that Michelle will have about $1,701 at the end of three years.

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Question

Jack has $\$15$ , to invest. If he invests two-thirds of it into a high-yield savings account with an annual interest rate of , compounded quarterly, and the other third in a regular savings account at simple interest, how much does Jack earn after one year?

Answer

First, break the problem into two segments: the amount Jack invests in the high-yield savings, and the amount Jack invests in the simple interest account (10,000 and 5,000 respectively).

Now let's work with the high-yield savings account. $10,000 is invested at an annual rate of 8%, compounded quarterly. We can use the compound interest formula to solve:

$\text{Final Balance}=\text{ Principal} \cdot \bigg(1+\frac{\text{Interest Rate}}{c}\bigg)^{(\text{Time})(c)}$

Plug in the values given:

$=10,000\cdot \bigg(1+\frac{.08}{4}\bigg)^{(1)(4)}$

$=10,000 \cdot (1.02)^{4}$

Therefore, Jack makes $824.32 off his high-yield savings account. Now let's calculate the other interest:

$\text{Interest}=\text{Principal} \cdot \text{ Interest Rate} \cdot \text{ Time}$

$=5,000 \cdot (0.06)\cdot (1)$

Add the two together, and we see that Jack makes a total of, $\$1124.32$ off of his investments.

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Question

Ashley makes a bank deposit of $\$2,500$ at annual interest, compounded monthly. About how many years will it take her deposit to grow to $\$3,500$ ?

Answer

The formula for compound interest is

$A = P\left ( 1 + \frac{r}{n}\right )^{nt}$

where P is the principal (original) amount, r is the interest rate (in decimal form), n is the number of times per year the interest compounds, t is the time in years, and A is the final amount.

In this problem, we are solving for time, t. The given variables from the problem are:

Plugging these into the equation above, we get

$3500 = 2500\left ( 1 + \frac{0.031}{12}\right )^{12t}$

This simplifies to

$1.4 = \left ( 1 + \frac{0.031}{12}\right )^{12t}$

We can solve this by taking the natural log of both sides

$\ln \left (1.4 \right ) = \ln \left (\left ( 1 + \frac{0.031}{12}\right )^{12t} \right )$

$\ln \left (1.4 \right ) = 12t\ln \left ( 1 + \frac{0.031}{12}\right )$

$\frac{\ln \left (1.4 \right ) }{\ln \left (1 + \frac{0.031}{12}\right ) }= 12t$

$\frac{\ln \left (1.4 \right ) }{12\ln \left (1 + \frac{0.031}{12}\right ) }= t$

$t = 10.86 \approx 11$

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