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Decide whether each statements is true or false.

$\left(7+8\right)^3=7^3+8^3$

$(xy)^3=x^3 y^3$

$2x^3 \cdot 4x^2 \cdot 5x^6=(2 \cdot 4 \cdot 5)x^{3+2+6}$

$\left(x^3y^5\right)^4=x^{3+4}y^{5+4}$

$2\left(x^2y^5\right)^3=8x^6y^{15}$

$x^2+x^3=x^5$

$x^3+x^3=2x^3$

$x^3\cdot x^3=2x^6$

$3^2 \cdot 2^3=6^5$

$3^{-2} =-\frac{1}{9}$

Solution

False, $\left(7+8\right)^3\neq 7^3+8^3\text{.}$ Following the order of operations, on the left $\left(7+8\right)^3$ would simplify as $15^3\text{,}$ which is $3375\text{.}$ However, on the right side, we have
$\begin{aligned} 7^3+8^3\amp=343+512\\ \amp=855 \end{aligned}$
Since $3375\neq 855\text{,}$ the equation is false.

True. As the cube applies to the product of $x$ and $y\text{,}$ $(xy)^3=x^3 y^3\text{.}$

True. The coefficients do get multiplied together and the exponents added when the expressions are multiplied, so $2x^3 \cdot 4x^2 \cdot 5x^6=(2 \cdot 4 \cdot 5)x^{3+2+6}\text{.}$

False, $\left(x^3y^5\right)^4\neq x^{3+4}y^{5+4}\text{.}$ When we have a power to a power, we multiply the exponents rather than adding them. So
$\left(x^3y^5\right)^4=x^{3 \cdot 4}y^{5 \cdot 4}$

False, $2\left(x^2y^5\right)^3\neq 8x^6y^{15}\text{.}$ The exponent of $3$ applies to $x^2$ and $y^{5}\text{,}$ but does not apply to the $2\text{.}$ So
$\begin{aligned} 2\left(x^2y^5\right)^3\amp=2x^{2\cdot 3}6y^{5\cdot 3}\\ \amp=2x^6y^{15} \end{aligned}$

False, $x^2+x^3\neq x^5\text{.}$ The two terms on the left hand side are not like terms and there is no way to combine them.

True. The terms $x^3$ and $x^3$ are like terms, so $x^3+x^3=2x^3\text{.}$

False, $x^3\cdot x^3 \neq 2x^6\text{.}$ When $x^3$ and $x^3$ are multiplied, their coefficients are each $1\text{.}$ So the coefficient of their product is still $1\text{,}$ and we have $x^3 \cdot x^3 = x^6\text{.}$

False, $3^2 \cdot 2^3\neq 6^5\text{.}$ Note that neither the bases nor the exponents are the same. Following the order of operations, on the left $3^2 \cdot 2^3$ would simplify as $9 \cdot 8\text{,}$ which is $72\text{.}$ However, on the right side, we have $6^5=7776\text{.}$ Since $72\neq 7776\text{,}$ the equation is false.

False, $3^{-2}\neq -\frac{1}{9}\text{.}$ The exponent of $-2$ on the number $3$ does not result in a negative number. Instead, $3^{-2}= \frac{1}{3^2}\text{,}$ which is $\frac{1}{9}\text{.}$