1
Which expression is equivalent to \(42p-54\)?
Equivalent expressions: worksheet with answers
Work through the questions first, then open the answer key and explanations to check your work.
2
Which expression is equivalent to \(7(n-6)\)?
3
Which expression is equivalent to \(14t^{5}-6t^{5}\)?
4
Which expression is equivalent to \(m^{2}-7m-30\)?
5
Which expression is equivalent to \((m^{-3}n^{4}p^{-1})(m^{5}p^{2}-n^{-2}p^{4})\)?
6
Which expression is equivalent to \((m^{-3}n^{4}p^{-2})(m^{5}p^{3}-n^{2}p^{-4})\)?
7
For positive real numbers \(a\), \(b\), and \(c\), which expression is equivalent to \(\sqrt[6]{a^{9}b^{9}c^{9}}\)?
8
The expression \((7m^{4}-9)-(-5m^{4}+6)\) is equivalent to \(km^{4}-15\). What is the value of \(k\)?
9
Which expression is equivalent to
\(\frac{3y+8}{x-4}+\frac{2y(x-4)}{y(x-4)(y+2)}\)?
10
Let \(c\) be a nonzero constant. If \((z-2c)\) is a factor of \(z^{2}-\frac{m}{4}c^{2}\), what is the value of \(m\)?
11
Which expression is equivalent to
\(\frac{3y+8}{x-4}+\frac{2y(x-4)}{y(x-4)(x+1)}\)?
12
Let \(u\) be a nonzero number. If \((z-4u)\) is a factor of \(z^{2}-\frac{p}{4}u^{2}\), what is the value of \(p\)?
Answer key and explanations
1. Answer
\(6(7p-9)\)
To factor \(42p-54\), find the greatest common factor of 42 and 54, which is 6. Dividing each term by 6 gives \(42p\div 6=7p\) and \(-54\div 6=-9\). Therefore, \(42p-54=6(7p-9)\). Expanding \(6(7p-9)\) gives \(42p-54\), so choice C is correct.
Choice A expands to \(252p-324\). Choice B expands to \(42p-48\). Choice D expands to \(48p-54\). None of these are equivalent to the original expression.
2. Answer
\(7n-42\)
Use the distributive property to multiply 7 by each term inside the parentheses: \(7(n-6)=7n-42\). Therefore, the equivalent expression is \(7n-42\).
3. Answer
\(8t^{5}\)
The two terms are like terms because they both contain \(t^{5}\). Subtract the coefficients: \(14-6=8\). The exponent stays the same, so the expression simplifies to \(8t^{5}\).
Choice A adds the exponents incorrectly. Choice B adds the coefficients instead of subtracting them. Choice D combines both the coefficients and exponents incorrectly.
4. Answer
\((m-10)(m+3)\)
To factor \(m^{2}-7m-30\), find two integers whose product is \(-30\) and whose sum is \(-7\). The integers are \(-10\) and \(3\) because \((-10)(3)=-30\) and \(-10+3=-7\). Therefore, the trinomial factors as \((m-10)(m+3)\).
5. Answer
\(m^{2}n^{4}p-m^{-3}n^{2}p^{3}\)
Distribute \(m^{-3}n^{4}p^{-1}\) to each term in the binomial.
First term: \((m^{-3}n^{4}p^{-1})(m^{5}p^{2})=m^{-3+5}n^{4}p^{-1+2}=m^{2}n^{4}p\).
Second term: \((m^{-3}n^{4}p^{-1})(-n^{-2}p^{4})=-m^{-3}n^{4+(-2)}p^{-1+4}=-m^{-3}n^{2}p^{3}\).
Therefore, the equivalent expression is \(m^{2}n^{4}p-m^{-3}n^{2}p^{3}\).
6. Answer
\(m^{2}n^{4}p-m^{-3}n^{6}p^{-6}\)
Distribute the monomial \(m^{-3}n^{4}p^{-2}\) to each term in the binomial.
First term: \((m^{-3}n^{4}p^{-2})(m^{5}p^{3})=m^{-3+5}n^{4}p^{-2+3}=m^{2}n^{4}p\).
Second term: \((m^{-3}n^{4}p^{-2})(-n^{2}p^{-4})=-m^{-3}n^{4+2}p^{-2+(-4)}=-m^{-3}n^{6}p^{-6}\).
So the equivalent expression is \(m^{2}n^{4}p-m^{-3}n^{6}p^{-6}\).
7. Answer
\((abc)^{\frac{3}{2}}\)
Because each factor inside the radical has the same exponent, first group the variables together:
\(\sqrt[6]{a^{9}b^{9}c^{9}}=\sqrt[6]{(abc)^{9}}\)
Next, rewrite the sixth root as a rational exponent of \(\frac{1}{6}\):
\(\left((abc)^{9}\right)^{\frac{1}{6}}\)
Apply the power-of-a-power rule by multiplying the exponents:
\((abc)^{\frac{9}{6}}=(abc)^{\frac{3}{2}}\)
So the correct answer is \((abc)^{\frac{3}{2}}\).
8. Answer
12
Distribute the subtraction across the second set of parentheses:
\((7m^{4}-9)-(-5m^{4}+6)=7m^{4}-9+5m^{4}-6\)
Combine like terms:
\(7m^{4}+5m^{4}=12m^{4}\) and \(-9-6=-15\).
So the expression simplifies to \(12m^{4}-15\). Since this is equivalent to \(km^{4}-15\), the value of \(k\) is 12.
9. Answer
\(\frac{3y^{2}+14y+2x+8}{xy+2x-4y-8}\)
First, factor the denominator of the second fraction:
\(\frac{2y(x-4)}{y(x-4)(y+2)}\)
The factors \(y\) and \(x-4\) cancel, leaving
\(\frac{2}{y+2}\)
Now the expression becomes
\(\frac{3y+8}{x-4}+\frac{2}{y+2}\)
The least common denominator is \((x-4)(y+2)\). Rewrite each fraction:
\(\frac{(3y+8)(y+2)}{(x-4)(y+2)}+\frac{2(x-4)}{(x-4)(y+2)}\)
Expand the numerators:
\((3y+8)(y+2)=3y^{2}+14y+16\)
\(2(x-4)=2x-8\)
Combine like terms:
\(3y^{2}+14y+16+2x-8=3y^{2}+14y+2x+8\)
So the equivalent expression is
\(\frac{3y^{2}+14y+2x+8}{xy+2x-4y-8}\)
because \((x-4)(y+2)=xy+2x-4y-8\).
10. Answer
16
If \((z-2c)\) is a factor of \(z^{2}-\frac{m}{4}c^{2}\), then substituting \(z=2c\) into the expression must give 0.
\((2c)^{2}-\frac{m}{4}c^{2}=0\)
\(4c^{2}-\frac{m}{4}c^{2}=0\)
Because \(c\neq 0\), divide both sides by \(c^{2}\):
\(4-\frac{m}{4}=0\)
\(\frac{m}{4}=4\), so \(m=16\).
Therefore, the correct answer is 16.
11. Answer
\(\frac{3xy+10x+3y}{x^{2}-3x-4}\)
First, factor the denominator of the second fraction:
\(\frac{2y(x-4)}{y(x-4)(x+1)}=\frac{2}{x+1}\)
The expression becomes
\(\frac{3y+8}{x-4}+\frac{2}{x+1}\).
The least common denominator is \((x-4)(x+1)\). Rewrite each fraction:
\(\frac{(3y+8)(x+1)}{(x-4)(x+1)}+\frac{2(x-4)}{(x-4)(x+1)}\)
Expand the numerators:
\((3y+8)(x+1)=3xy+8x+3y+8\)
\(2(x-4)=2x-8\)
Combine like terms:
\(3xy+8x+3y+8+2x-8=3xy+10x+3y\)
The equivalent expression is
\(\frac{3xy+10x+3y}{x^{2}-3x-4}\), since \((x-4)(x+1)=x^{2}-3x-4\).
12. Answer
64
If \((z-4u)\) is a factor of \(z^{2}-\frac{p}{4}u^{2}\), then the expression must match a difference of squares.
Since \(z^{2}-(4u)^{2}=z^{2}-16u^{2}\), the expression can be written as \((z-4u)(z+4u)\).
Therefore, the coefficient of \(u^{2}\) in the original expression must satisfy
\(\frac{p}{4}=16\).
Multiplying both sides by 4 gives \(p=64\).