Sure, I'll solve each of the given questions step-by-step.
1. [tex]\(a \times a \times a\)[/tex]
When multiplying the same base number, you add the exponents. Here, we have [tex]\(a\)[/tex] multiplied by itself three times:
[tex]\[ a \times a \times a = a^3 \][/tex]
2. [tex]\(3a \times 3a \times 3a\)[/tex]
Again, when multiplying numbers and variables, you multiply the coefficients (numbers) together and then add the exponents of the variables:
[tex]\[ (3a) \times (3a) \times (3a) = 3 \times 3 \times 3 \times a \times a \times a = 27a^3 \][/tex]
3. [tex]\(a^2 \times a^3\)[/tex]
When multiplying terms with the same base, you add the exponents:
[tex]\[ a^2 \times a^3 = a^{2+3} = a^5 \][/tex]
4. [tex]\(4a^2 \cdot 5a^b\)[/tex]
First, multiply the coefficients (4 and 5), then add the exponents of the common bases (since both terms have [tex]\(a\)[/tex]):
[tex]\[ 4a^2 \cdot 5a^b = 4 \times 5 \times a^{2+b} = 20a^{2+b} \][/tex]
b. [tex]\(\left(-5x^2\right)(3x)\)[/tex]
Multiply the coefficients (-5 and 3) and add the exponents of [tex]\(x\)[/tex]:
[tex]\[ (-5x^2)(3x) = -5 \times 3 \times x^{2+1} = -15x^3 \][/tex]
v. [tex]\(\left(6b^5\right)\left(-2ab^4\right)\)[/tex]
Again, multiply the coefficients (6 and -2), and add the exponents of the common bases (here, [tex]\(b\)[/tex]):
[tex]\[ (6b^5)(-2ab^4) = 6 \times -2 \times a \times b^{5+4} = -12ab^9 \][/tex]
8. [tex]\(11m^3n \times (-2m^3)\)[/tex]
Multiply the coefficients (11 and -2), and add the exponents of the common bases (here, [tex]\(m\)[/tex]):
[tex]\[ 11m^3n \times (-2m^3) = 11 \times -2 \times m^{3+3} \times n = -22m^6n \][/tex]
So, the final answers are:
1. [tex]\(a^3\)[/tex]
2. [tex]\(27a^3\)[/tex]
3. [tex]\(a^5\)[/tex]
4. [tex]\(20a^{2+b}\)[/tex]
b. [tex]\(-15x^3\)[/tex]
v. [tex]\(-12ab^9\)[/tex]
8. [tex]\(-22m^6n\)[/tex]
