How to multiply exponential variables - ISEE Upper Level Mathematics Achievement

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Question

Simplify the following:

$z^{2}\times z^{5}$

Answer

To multiply variables with exponents, add the exponents. So,

$z^{2}\times z^{5}=z^{7}$

A longer way would be to write out all the multiplies the exponent tells us to do. This is a little clearer on why adding the exponents works but takes longer and isn't necessary once you understand the process.

$z^{2}\times z^{5}= zzzzzzz= z^{7}$

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Question

Simplify the following:

$(z^{3} x^{2} y)(x^{3} y^{4})$

Answer

To multiply variables with exponents, add the exponents. With multiple variables, simply add the exponents for each different variable.

Simplified:

$(z^{3})(x^{2+3})(y^{1+4})$

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Question

Simplify the following:

$(3py^{2})(4p^{4})$

Answer

To multiply variables with exponents, add the exponents. When there are constants mixed in, multiply the constants separately and put back in the final result:

$(3py^{2})(4p^{4})$

$3\times4\times p^{1+4}\times y^{2}$

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Question

Factor completely:

$x^{3} - 3x^{2}+ 4x$

Answer

is the greatest common factor of each term, so distribute it out:

$x^{3} - 3x^{2}+ 4x$

$= x \left (\frac{ x^{3} }{x}- \frac{3x^{2} }{x}+ \frac{4x}{x} \right )$

$= x \left (x^{2}- 3x + 4 \right )$

We try to factor $x^{2}- 3x + 4$ by finding two integers with product 4 and sum . However, both of our possible factor pairs fail, since and .

$x \left (x^{2}- 3x + 4 \right )$ is the complete factorization.

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Question

Factor completely:

$4 y^{2} - 100 y$

Answer

The greatest common factor of the terms in $4 y^{2} - 100 y$ is , so factor that out:

$4 y^{2} - 100 y$

$= 4y\left ( \frac{4 y^{2}}{4y} - \frac{100 y}{4y} \right )$

$= 4y\left (y - 25 \right )$

Since all factors here are linear, this is the complete factorization.

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Question

Multiply:

$\left ( x -3y + 1\right ) \left ( x - 3y - 1\right )$

Answer

This can be achieved by using the pattern of difference of squares:

$\left ( x -3y + 1\right ) \left ( x - 3y - 1\right )$

$=\left [ \left ( x -3y \right ) + 1\right ] \left \left[( x - 3y \right ) - 1\right ]$

$= ( x -3y ) ^{2} - 1 ^{2}$

$= (x -3y) ^{2} - 1$

Applying the binomial square pattern:

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

$= x^{2} - 6xy +9y ^{2} - 1$

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Question

Multiply:

Answer

Use the distributive property, then collect like terms:

$= (x + y + z) \cdot x + (x + y + z) \cdot y$

$= x\cdot x + y \cdot x + z\cdot x + x \cdot y+ y \cdot y+ z \cdot y$

$= x^{2}+ xy+ xz + x y+ y^{2}+ yz$

$= x^{2}+ xy+ x y + yz + xz + y^{2}$

$= x^{2}+2 xy+ yz + xz + y^{2}$

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Question

Exponentiate:

$\left (x + 0.5y \right) ^{3}$

Answer

The cube of a sum pattern can be applied here:

$\left (x + 0.5y \right) ^{3}$

$= x^{3} + 3 \cdot x^{2}\cdot 0.5y + 3 \cdot x \cdot\left (0.5y \right) ^{2} + \left (0.5y \right) ^{3}$

$= x^{3} + 3\cdot 0.5 \cdot x^{2} \cdot y + 3 \cdot 0.5^{2} \cdot x \cdot y ^{2} + 0.5^{3} \cdot y ^{3}$

$= x^{3} + 3 \cdot 0.5 x^{2} y + 3 \cdot 0.25 \ x y ^{2} + 0.125 y ^{3}$

$= x^{3} + 1.5 x^{2} y + 0.75 \ x y ^{2} + 0.125 y ^{3}$

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Question

Write in expanded form:

$\left (2x + \frac{1}{2} \right )^{3}$

Answer

The cube of a sum pattern can be applied here:

$\left (2x + \frac{1}{2} \right )^{3}$

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

$= 2^{3}\cdot x^{3} + 3\cdot 2^{2}\cdot \frac{1}{2} \cdot x^{2} + 3\cdot2 \cdot \left (\frac{1}{2} \right ) ^{2} \cdot x \left + \left ( \frac{1}{2} \right ) ^{3}$

$= 8x^{3} + 6 x^{2} + \frac{3}{2} x + \frac{1}{8}$

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Question

Factor completely:

$x ^{2} - 11x + 20$

Answer

A trinomial with leading coefficient $x^{2}$ can be factored by looking for two integers to fill in the boxes:

$(x + \square )(x + \square )$ .

The numbers should have product 20 and sum . However, all of the possible factor pairs fail:

The polynomial is prime.

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Question

Factor completely:

$4x^{2} - 36x+ 81$

Answer

$4x^{2} - 36x+ 81$ can be seen to be a perfect square trinomial by taking the square root of the first and last terms, multiplying their product by 2, then comparing it to the second term:

$4x^{2} = \left ( 2x\right )^{2}$

$81 = 9^{2}$

$2 \cdot2x\cdot9= 36x$

Therefore,

$4x^{2} - 36x+ 81$

$=\left ( 2x \right ) ^{2} -2 \cdot 2 x \cdot9+ 9 ^{2}$

$=(2x-9)^{2}$

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Question

Fill in the box to form a perfect square trinomial:

$x ^{2} + 20x + \square$

Answer

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is 20, by 2, and square the quotient. The result is

$\left (\frac{20}{2} \right )^{2} =10 ^{2} =100$

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Question

Fill in the box to form a perfect square trinomial:

$x ^{2} + 9x + \square$

Answer

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is 9, by 2, and square the quotient. The result is

$\left (\frac{9}{2} \right) ^{2} = \frac{9^{2}}{2^{2}}= \frac{81}{4}$

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Question

Multiply:

Answer

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Question

Simplify:

$(0.5x-0.4)^{2}$

Answer

Use the pattern, substituting .

$(A - B)^{2} = A^{2} - 2AB + B^{2}$

$(0.5x - 0.4)^{2} = 0.5x^{2} - 2\cdot 0.5x \cdot 0.4+ 0.4^{2}$

$= 0.25x^{2} - 0.4x+ 0.16$

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Question

Write in expanded form.

$(4x+\frac{3}{x})^3$

Answer

$(4x+\frac{3}{x})^3$

$=(4x)^3+3(4x)^2(\frac{3}{x})+3(4x)(\frac{3}{x})^2+(\frac{3}{x})^3$

$=4^3\times x^3+3\times 4^2\times 3\times \frac{x^2}{x}+3\times 4\times 3^2\times \frac{x}{x^2}+\frac{3^3}{x^3}$

$=64x^3+144x+\frac{108}{x}+\frac{27}{x^3}$

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Question

Simplify:

Answer

First, simplify all of the exponents. When the exponent is outside of the parantheses, multiply it by the exponents inside so that you get: . Multiply so that you get 27. Then, multiply like terms. First, multilpy 2 by 27 so that you get 54. Then, multiply the x terms. Remember, when bases are the same, add the exponents: $x^4\cdot x^3=x^7$ . Then, multiply the y terms: $y^6\cdot y^6=y^{12}$ . Then, multiply all of the terms together so that you get $54x^7y^{12}$ .

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Question

Simplify:

$\frac{4x^{3}}{y} \cdot \frac{y^{5}}{8x} \cdot xy$

Answer

$\frac{4x^{3}}{y} \cdot \frac{y^{5}}{8x} \cdot xy$

$= \frac{4x^{3}}{y} \cdot \frac{y^{5}}{8x} \cdot \frac{xy}{1}$

$= \frac{4x^{3} \cdot y^{5} \cdot xy }{y \cdot 8x \cdot 1 }$

$= \frac{4x^{3+1 } \cdot y^{5+1} }{8xy }$

$= \frac{4x^{4 } \cdot y^{6} }{8xy }$

$= \frac{ x^{4-1 } \cdot y^{6-1} }{2 }$

$= \frac{ x^{3 } y^{5} }{2 }$

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Question

Simplify the following expression:

Answer

Simplify the following expression:

To combine these, we need to multiply our coefficients and our variables.

First, multiply the coefficients

Next, multiply our variables by adding the exponent:

$(x^5y^2)(x^3y^8)=x^{5+3}y^{2+8}=x^8y^{10}$

So, we put it all together to get:

$108x^8y^{10}$

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Question

Simplify the following:

$a^2 \cdot a^4$

Answer

When multiply variables with exponents, we will use the following formula:

$a^x \cdot a^y = a^{x+y}$

So, we can write the problem like this:

$a^2 \cdot a^4 = a^{2+4}$

$a^2 \cdot a^4 = a^6$

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