Question 1

Use the distributive property to write the expression $x ( x + 8 )$ without parentheses. Simplify the result, if possible.
A) $x ^ { 2 } + 8 x$
B) $x + 8 x$
C) $x + x + 8 x$
D) $2 x + 8$
E) $x ^ { 2 } + 8 x ^ { 2 } + 8 x$

Accepted Answer

Using the distributive property, we have:

$$x(x+8)=x\cdot x + x \cdot 8 = x^2+8x$$

There is no further simplification possible.

Question 2

Evaluate the following expression for the given value of the variable. $\frac { z - 10 } { 10 - z }$  for $z = 8$
A) 1
B) -2
C) $- 1$
D) 2
E) 0

Accepted Answer

The answer of Evaluate the following expression for the given...

Question 3

Simplify and complete the expression. 
5( y - 7) + 8 y = 5 y +      - 35 =      - 35

Accepted Answer

The answer of Simplify and complete the expression. 
5( y...

Question 4

Determine the correct statement.
A) $a \cdot ( b + c ) = a \cdot b + c$
B) $a \cdot ( b + c ) = a \cdot b + a \cdot c$
C) $a \cdot ( b + c ) = a \cdot b + a$
D) $a \cdot ( b + c ) = a \cdot b + c \cdot b$
E) $a \cdot ( b + c ) = a \cdot b + \frac { 1 } { b }$

Accepted Answer

The distributive property of multiplication states that $a \cdot (b+c) = a \cdot b + a \cdot c$. This means that we multiply the factor outside the parentheses (a) by each term inside the parentheses (b and c), and then add those two products together. Choice B is the only option that correctly uses the distributive property.

Question 5

Simplify the expression.  $\left( 3 s ^ { 3 } \right) ^ { 3 }$
A) $27 s ^ { 6 }$
B) $27 s ^ { 9 }$
C) $3 s ^ { 9 }$
D) $27 s ^ { 3 }$
E) $3 s ^ { 3 }$

Accepted Answer

To simplify, we can multiply the exponents: 
$\left( 3 s ^ { 3 } \right) ^ { 3 } = 3 ^ 3 s ^ {3 \cdot 3} = 27 s ^ 9$

Question 6

Use the distributive property to write the expression $8 x ( d - x )$  without parentheses. Simplify the result, if possible.
A) $8 d x - 8 d ^ { 2 }$
B) $d x + d - 8 x ^ { 2 }$
C) $8 x + d - x$
D) $8 d x - 8 x ^ { 2 }$
E) $d x - 8 x ^ { 2 }$

Accepted Answer

Using the distributive property, we get:
$$8 x ( d - x ) = 8 x d - 8 x^2 = 8 dx - 8x^2$$
which matches option D. We cannot simplify this expression any further.

Question 7

Find the product. $\left( 6 t ^ { 4 } x ^ { 4 } \right) \left( 5 t ^ { 2 } x ^ { 2 } \right)$
A) $11 t ^ { 6 } x ^ { 6 }$
B) $30 t ^ { 6 } x ^ { 6 }$
C) $30 t ^ { 8 } x ^ { 8 }$
D) $30 t ^ { 2 } x ^ { 2 }$
E) none of these

Accepted Answer

To multiply two expressions together, we multiply the coefficients (numbers in front) and then add the exponents of each variable. Applying this rule, we get:
\begin{align*} (6t^4x^4)(5t^2x^2)&=30t^{4+2}x^{4+2} \ &=30t^6x^6\end{align*}
Therefore, the answer is (B) 30t^6x^6.

Question 8

Select the expression that completes the formula below. $( x y ) ^ { k }$
A) $x ^ { k } y ^ { k }$
B) $x ^ { m + k }$
C) $x ^ { m k }$
D) $\frac { x ^ { k } } { y ^ { k } }$
E) none of these

Accepted Answer

To raise a product to a power, we raise each factor to the power. Therefore, $(xy)^k$ = $x^ky^k$.

Question 9

Let $y = 5$ and $z = 1$ . Write the phrase as an algebraic expression, and evaluate it. $y$ minus $Z$
A) $y - z$ ; $5$
B) $y - z$ ; $4$
C) $y + z$ ; $5$
D) $y + z$ ; $4$
E) $y - z$ ; $- 4$

Accepted Answer

The phrase "y minus z" translates to the algebraic expression $y - z$. Substituting the given values, $5 - 1 = 4$.

Question 10

Evaluate the expression. $6 \cdot 10 ^ { 3 } + 5 \cdot 10 ^ { 2 } + 4 \cdot 10 ^ { 1 } + 8$
A) 6,648
B) 6,548
C) 6,553
D) 6,537
E) 6,546

Accepted Answer

To evaluate this expression, we use the rules of exponents to rewrite it as $6	imes10^3+5	imes10^2+4	imes10^1+8=6000+500+40+8=6548$. Therefore, the best choice is (B).

Question 11

Find the product. $\left( 7 y ^ { 6 } x ^ { 6 } \right) \left( 2 y ^ { 2 } x ^ { 4 } \right)$
A) $16 y ^ { 9 } x ^ { 11 }$
B) $14 y ^ { 8 } x ^ { 10 }$
C) $14 y ^ { 4 } x ^ { 2 }$
D) $14 y ^ { 12 } x ^ { 24 }$
E) $9 y ^ { 8 } x ^ { 10 }$

Accepted Answer

The product is found by multiplying the coefficients and adding the exponents of like bases: $7 \times 2 = 14$, $y^{6+2} = y^8$, $x^{6+4} = x^{10}$. Thus, the product is $14 y^8 x^{10}$.

Question 12

Simplify the following algebraic expression and express your answer using positive exponents only. $\left( a ^ { - 1 } v ^ { - 2 } \right) ^ { 7 }$
A) $a ^ { 8 } v ^ { 9 }$
B) $\frac { 1 } { a ^ { 8 } v ^ { 9 } }$
C) $\frac { 1 } { a ^ { 6 } v ^ { 5 } }$
D) $a ^ { 7 } v ^ { 14 }$
E) $\frac { 1 } { a ^ { 7 } v ^ { 14 } }$

Accepted Answer

When raising a power to a power, multiply the exponents: $(a^{-1})^7 = a^{-7}$ and $(v^{-2})^7 = v^{-14}$. Thus, $(a^{-1}v^{-2})^7 = a^{-7}v^{-14}$. To express with positive exponents, take the reciprocal: $\frac{1}{a^7v^{14}}$.

Question 13

Simplify the expression. $\left( 6 t ^ { 3 } \right) ^ { 2 }$
A) $6 t ^ { 6 }$
B) $6 t ^ { 3 }$
C) $36 t ^ { 7 }$
D) $36 t ^ { 6 }$
E) $36 t ^ { 5 }$

Accepted Answer

When raising a power to a power, you multiply the exponents. Therefore, $\left( 6 t ^ { 3 } \right) ^ { 2 } = 6 ^ 2 \cdot t ^ { 3 \cdot 2 } = 36 t ^ 6$

Question 14

What is $t ^ { 3 } \cdot t ^ { 3 }$  rewritten using only one exponent?
A) $t ^ { 0 }$
B) $t ^ { 6 }$
C) $t ^ { 3 }$
D) $t ^ { 9 }$
E) none of these

Accepted Answer

When we multiply two numbers with the same base, we add their exponents. Here, the base is "t" and we are multiplying two t's, so we add their exponents, which are both 3. Thus, the answer is t to the power of 6, written as t^6.

Question 15

Simplify the product. $- 3 x ^ { 2 } c ^ { 5 } \left( - 2 c ^ { 2 } x ^ { 5 } \right)$
A) $6 x ^ { 7 } c ^ { 7 }$
B) $- 6 x ^ { 5 } c ^ { 7 }$
C) $2 x ^ { 7 } c ^ { 7 }$
D) $3 x ^ { 7 } c ^ { 2 }$
E) none of these

Accepted Answer

The answer of Simplify the product. \(- 3 x ^...

Question 16

Write the following statement as a mathematical expression.
Three is less than twenty-three.
A) $3 \geq 23$
B) $3 = 23$
C) $3 \leq 23$
D) $3 < 23$
E) $3 > 23$

Accepted Answer

"Less than" translates to the symbol "<" in mathematical expressions. Therefore, the correct expression is $3 < 23$.

Question 17

Evaluate the following expression for the given value of the variable. $\frac { y - 4 } { 5 }$  for $y = 14$
A) $- 10$
B) $6$
C) $10$
D) $2$
E) $- 2$

Accepted Answer

The answer of Evaluate the following expression for the given...

Question 18

Let x , y , and z represent three real numbers. Write an algebraic expression to denote the product of z , x , and y .
A) $x y - z$
B) $z - y$
C) $x y z$
D) $z - x y$
E) $x + 2 z$

Accepted Answer

The product of z, x, and y is represented by the expression xyz.

Question 19

Find the product. $\left( 6 b ^ { 5 } \right) \left( 5 b ^ { 6 } \right)$
A) $30 b ^ { 11 }$
B) $30 b ^ { 30 }$
C) $11 b ^ { 11 }$
D) $11 b ^ { 30 }$
E) none of these

Accepted Answer

The product of the coefficients is $6 \times 5 = 30$. The product of the powers of $b$ is $b^5 \times b^6 = b^{5+6} = b^{11}$. Thus, the product is $30 b^{11}$.

Question 20

What is $2 c ^ { 3 } \left( 4 c ^ { 2 } \right) 5 c ^ { 2 }$  rewritten using only one exponent?
A) $8 c ^ { 7 }$
B) $40 c$
C) $40 c ^ { 6 }$
D) $40 c ^ { 7 }$
E) $9 c ^ { 7 }$

Accepted Answer

When we multiply terms with the same base, we add their exponents. Applying this rule, we have:

\begin{align*}
2c^3(4c^2)5c^2 &= 2\cdot4\cdot5\cdot c^3\cdot c^2\cdot c^2\
&= 40c^{3+2+2}\
&= 40c^7
\end{align*}

Therefore, the answer is D), $40c^7$.

Quiz 8: An Introduction to Algebra

Use the distributive property to write the expression x(x+8) x ( x + 8 ) x(x+8) without parentheses. Simplify the result, if possible.

Evaluate the following expression for the given value of the variable. z−1010−z\frac { z - 10 } { 10 - z }10−zz−10​ for z=8z = 8z=8

Simplify and complete the expression. 5( y - 7) + 8 y = 5 y + - 35 = - 35

Determine the correct statement.

Simplify the expression. (3s3) 3\left( 3 s ^ { 3 } \right) ^ { 3 }(3s3) 3

Use the distributive property to write the expression 8x(d−x) 8 x ( d - x ) 8x(d−x) without parentheses. Simplify the result, if possible.

Find the product. (6t4x4) (5t2x2) \left( 6 t ^ { 4 } x ^ { 4 } \right) \left( 5 t ^ { 2 } x ^ { 2 } \right) (6t4x4) (5t2x2)

Select the expression that completes the formula below. (xy) k( x y ) ^ { k }(xy) k

Let y=5y = 5y=5 and z=1z = 1z=1 . Write the phrase as an algebraic expression, and evaluate it. yyy minus ZZZ

Evaluate the expression. 6⋅103+5⋅102+4⋅101+86 \cdot 10 ^ { 3 } + 5 \cdot 10 ^ { 2 } + 4 \cdot 10 ^ { 1 } + 86⋅103+5⋅102+4⋅101+8

Find the product. (7y6x6) (2y2x4) \left( 7 y ^ { 6 } x ^ { 6 } \right) \left( 2 y ^ { 2 } x ^ { 4 } \right) (7y6x6) (2y2x4)

Simplify the following algebraic expression and express your answer using positive exponents only. (a−1v−2) 7\left( a ^ { - 1 } v ^ { - 2 } \right) ^ { 7 }(a−1v−2) 7

Simplify the expression. (6t3) 2\left( 6 t ^ { 3 } \right) ^ { 2 }(6t3) 2

What is t3⋅t3t ^ { 3 } \cdot t ^ { 3 }t3⋅t3 rewritten using only one exponent?

Simplify the product. −3x2c5(−2c2x5) - 3 x ^ { 2 } c ^ { 5 } \left( - 2 c ^ { 2 } x ^ { 5 } \right) −3x2c5(−2c2x5)

Write the following statement as a mathematical expression. Three is less than twenty-three.

Evaluate the following expression for the given value of the variable. y−45\frac { y - 4 } { 5 }5y−4​ for y=14y = 14y=14

Let x , y , and z represent three real numbers. Write an algebraic expression to denote the product of z , x , and y .

Find the product. (6b5) (5b6) \left( 6 b ^ { 5 } \right) \left( 5 b ^ { 6 } \right) (6b5) (5b6)

What is 2c3(4c2) 5c22 c ^ { 3 } \left( 4 c ^ { 2 } \right) 5 c ^ { 2 }2c3(4c2) 5c2 rewritten using only one exponent?

Use the distributive property to write the expression $x ( x + 8 )$ without parentheses. Simplify the result, if possible.

Evaluate the following expression for the given value of the variable. $\frac { z - 10 } { 10 - z }$ for $z = 8$

Simplify the expression. $\left( 3 s ^ { 3 } \right) ^ { 3 }$

Use the distributive property to write the expression $8 x ( d - x )$ without parentheses. Simplify the result, if possible.

Find the product. $\left( 6 t ^ { 4 } x ^ { 4 } \right) \left( 5 t ^ { 2 } x ^ { 2 } \right)$

Select the expression that completes the formula below. $( x y ) ^ { k }$

Let $y = 5$ and $z = 1$ . Write the phrase as an algebraic expression, and evaluate it. $y$ minus $Z$

Evaluate the expression. $6 \cdot 10 ^ { 3 } + 5 \cdot 10 ^ { 2 } + 4 \cdot 10 ^ { 1 } + 8$

Find the product. $\left( 7 y ^ { 6 } x ^ { 6 } \right) \left( 2 y ^ { 2 } x ^ { 4 } \right)$

Simplify the following algebraic expression and express your answer using positive exponents only. $\left( a ^ { - 1 } v ^ { - 2 } \right) ^ { 7 }$

Simplify the expression. $\left( 6 t ^ { 3 } \right) ^ { 2 }$

What is $t ^ { 3 } \cdot t ^ { 3 }$ rewritten using only one exponent?

Simplify the product. $- 3 x ^ { 2 } c ^ { 5 } \left( - 2 c ^ { 2 } x ^ { 5 } \right)$

Evaluate the following expression for the given value of the variable. $\frac { y - 4 } { 5 }$ for $y = 14$

Find the product. $\left( 6 b ^ { 5 } \right) \left( 5 b ^ { 6 } \right)$

What is $2 c ^ { 3 } \left( 4 c ^ { 2 } \right) 5 c ^ { 2 }$ rewritten using only one exponent?