Questions: a. The expression x^2 x^6 y^10 / x^3 y^4 can be rewritten in the form x^a y^b where: - a= - b= b. Which of the following expressions are equivalent to 2^4 * 5^3 ? Select all that apply 2^4 * 2 * 5^5 / 5^2 2^6 * 5^9 / 2^2 * 5^6 (2 * 5^2)(2^3 * 5) 2^4 * 5^2 * 5^6 / 5^5 2^7 * 5^3 / 2^3 * 5 (2^2 * 5)(2^2 * 5^3)

Solution Steps

a. To simplify the expression \(\frac{x^{2} x^{6} y^{10}}{x^{3} y^{4}}\), combine the exponents of like bases in the numerator and then subtract the exponents of the denominator from the numerator for each base.

b. For each expression, simplify it by applying the laws of exponents and check if it equals \(2^{4} \cdot 5^{3}\).

We start with the expression

\[ \frac{x^{2} x^{6} y^{10}}{x^{3} y^{4}}. \]

First, we combine the exponents in the numerator:

\[ x^{2} x^{6} = x^{2+6} = x^{8}. \]

Now, substituting this back into the expression gives us:

\[ \frac{x^{8} y^{10}}{x^{3} y^{4}}. \]

Next, we simplify by subtracting the exponents of like bases:

\[ \frac{x^{8}}{x^{3}} = x^{8-3} = x^{5}, \] \[ \frac{y^{10}}{y^{4}} = y^{10-4} = y^{6}. \]

Thus, the simplified expression is

\[ x^{5} y^{6}. \]

We need to determine which of the following expressions are equivalent to \(2^{4} \cdot 5^{3}\):

1. \(\frac{2^{4} \cdot 2 \cdot 5^{5}}{5^{2}}\)
2. \(\frac{2^{6} \cdot 5^{9}}{2^{2} \cdot 5^{6}}\)
3. \((2 \cdot 5^{2})(2^{3} \cdot 5)\)
4. \(\frac{2^{4} \cdot 5^{2} \cdot 5^{6}}{5^{5}}\)
5. \(9\)
6. \(\frac{2^{7} \cdot 5^{3}}{2^{3} \cdot 5}\)
7. \((2^{2} \cdot 5)(2^{2} \cdot 5^{3})\)

After simplifying each expression, we find that the following are equivalent to \(2^{4} \cdot 5^{3} = 2000\):

• Expression 2: \(2000\)
• Expression 3: \(2000\)
• Expression 4: \(2000\)

Final Answer