Apply the exponent rule b^0 = 1.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
kamogelo, let's knock this out.
a)
Step 1: Apply the exponent rule $b^0 = 1$.
$$3ab^0 \times 2a^2b = 3a(1) \times 2a^2b$$
Step 2: Multiply the coefficients.
$$3 \times 2 = 6$$
Step 3: Multiply the $a$ terms using the rule $a^m \times a^n = a^{m+n}$.
$$a \times a^2 = a^{1+2} = a^3$$
Step 4: Multiply the $b$ terms.
$$1 \times b = b$$
Step 5: Combine the results.
$$6a^3b$$
The simplified expression is:
$$\boxed{6a^3b}$$
b)
Step 1: Distribute $-x$ to each term inside the parentheses.
$$-x(2x+7) = (-x)(2x) + (-x)(7)$$
Step 2: Perform the multiplications.
$$-2x^2 - 7x$$
The simplified expression is:
$$\boxed{-2x^2 - 7x}$$
c)
Step 1: Distribute $3x^2$ to each term inside the parentheses.
$$3x^2(x^2+2) = (3x^2)(x^2) + (3x^2)(2)$$
Step 2: Perform the multiplications using the rule $x^m \times x^n = x^{m+n}$.
$$3x^{2+2} + 6x^2$$
$$3x^4 + 6x^2$$
The simplified expression is:
$$\boxed{3x^4 + 6x^2}$$
d)
Step 1: Distribute $-5$ to each term inside the parentheses.
$$-5(2x-6y+4) = (-5)(2x) + (-5)(-6y) + (-5)(4)$$
Step 2: Perform the multiplications.
$$-10x + 30y - 20$$
The simplified expression is:
$$\boxed{-10x + 30y - 20}$$
e)
Step 1: Apply the exponent rule $p^0 = 1$ inside the parentheses.
$$7p(-2p^0+4p-1) = 7p(-2(1)+4p-1)$$
$$7p(-2+4p-1)$$
Step 2: Distribute $7p$ to each term inside the parentheses.
$$7p(-2) + 7p(4p) + 7p(-1)$$
Step 3: Perform the multiplications.
$$-14p + 28p^2 - 7p$$
Step 4: Combine like terms.
$$28p^2 + (-14p - 7p)$$
$$28p^2 - 21p$$
The simplified expression is:
$$\boxed{28p^2 - 21p}$$
f)
Step 1: Distribute $4y$ to each term inside the parentheses.
$$4y(2y-3x+6) = (4y)(2y) + (4y)(-3x) + (4y)(6)$$
Step 2: Perform the multiplications.
$$8y^2 - 12xy + 24y$$
The simplified expression is:
$$\boxed{8y^2 - 12xy + 24y}$$
g)
Step 1: Distribute $3m$ into the first set of parentheses.
$$3m(-4m-3n) = (3m)(-4m) + (3m)(-3n)$$
$$-12m^2 - 9mn$$
Step 2: Distribute $-2m$ into the second set of parentheses.
$$-2m(-b+n) = (-2m)(-b) + (-2m)(n)$$
$$2mb - 2mn$$
Step 3: Combine the results from Step 1 and Step 2.
$$(-12m^2 - 9mn) - (2mb - 2mn)$$
Step 4: Distribute the negative sign to the terms in the second parenthesis.
$$-12m^2 - 9mn - 2mb + 2mn$$
Step 5: Combine like terms ($-9mn$ and $2mn$).
$$-12m^2 + (-9mn + 2mn) - 2mb$$
$$-12m^2 - 7mn - 2mb$$
The simplified expression is:
$$\boxed{-12m^2 - 7mn - 2mb}$$
h)
Step 1: Distribute $4x$ into the first set of parentheses.
$$4x(x+2) = (4x)(x) + (4x)(2)$$
$$4x^2 + 8x$$
Step 2: Distribute $-3x$ into the second set of parentheses.
$$-3x(x-4) = (-3x)(x) + (-3x)(-4)$$
$$-3x^2 + 12x$$
Step 3: Combine the results from Step 1 and Step 2.
$$(4x^2 + 8x) + (-3x^2 + 12x)$$
Step 4: Remove the parentheses.
$$4x^2 + 8x - 3x^2 + 12x$$
Step 5: Combine like terms ($4x^2$ with $-3x^2$, and $8x$ with $12x$).
$$(4x^2 - 3x^2) + (8x + 12x)$$
$$x^2 + 20x$$
The simplified expression is:
$$\boxed{x^2 + 20x}$$
What's next?
