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Elanso [62]
2 years ago
8

Find the least common denominator for these two rational expressions. B/b^2 -64 -7b/b^2+7b-8

Mathematics
1 answer:
san4es73 [151]2 years ago
3 0

Answer:

The least common denominator is (b-8)(b+8)(b-1)

Step-by-step explanation:

We are given expression as

\frac{b}{b^2-64}-\frac{7b}{b^2+7b-8}

Firstly, we will factor both denominators

b^2-64=b^2-8^2=(b-8)(b+8)

b^2+7b-8=(b+8)(b-1)

so, we can plug it back

\frac{b}{(b-8)(b+8)}-\frac{7b}{(b+8)(b-1)}

First term denominator is

(b-8)(b+8)

Second term denominator is

(b+8)(b-1)

So,

Least common denominator will be

(b-8)(b+8)(b-1)

So, we get

LCD=(x-8)(x+8)(x-1)


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Mr. Collins made 20 bird houses to sell at a carnival. He earns $15 for each bird house he sells. How much will Mr. Collins earn
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Answer:

C. $300

Step-by-step explanation:

We have been given that Mr. Collins made 20 bird houses to sell at a carnival. He earns $15 for each bird house he sells.

To find the amount earned by Mr. Collin after selling all 20 bird houses, we need to multiply 20 by amount earned by selling each bird house.

\text{Amount earned by Mr. Collin after selling all 20 bird houses}=\$15\times 20

\text{Amount earned by Mr. Collin after selling all 20 bird houses}=\$300

Therefore, Mr. Collin will earn $300, if he sells all 20 bird houses.

4 0
2 years ago
The symbol ⊕ is used to denote exclusive or, so p ⊕ q ≡ (p v q) ^ ~ (p ^ q).
Elan Coil [88]

I'll use text forms of the logical symbols because copying/pasting on mobile is annoying.

⊕ = XOR

∨ = OR

∧ = AND

~ = NOT

↔ = IFF (meaning "if and only if")

→ = IMP (short for "implies")

and for logical equivalence I'll use a regular equal sign.

Then

p XOR q = (p OR q) AND NOT (p AND q)

(a) By definition of XOR, we have

p XOR p = (p OR p) AND NOT (p AND p)

= p AND NOT p

and this is tautologically FALSE.

(b) To rewrite XOR, first distribute the NOT over the AND using DeMorgan's law:

NOT (p AND q) = NOT p OR NOT q

Then we can interchange the AND and ORs using their respective distributive properties. That is,

p XOR q = (p OR q) AND (NOT p OR NOT q)

= (p AND (NOT p OR NOT q)) OR (q AND (NOT p OR NOT q))

= ((p AND NOT p) OR (p AND NOT q)) OR ((q AND NOT p) OR (q AND NOT q))

Both (p AND NOT p) and (q AND NOT q) are tautologically FALSE, so the truth value of either OR depends only on the other statements, so we have

p XOR q = (p AND NOT q) OR (q AND NOT p)

Recall the definition of IFF:

p IFF q = (p IMP q) AND (q IMP p)

Also recall that IMP is logically equivalent to

p IMP q = p OR NOT q

Then we have

p IFF q = (p OR NOT q) AND (q OR NOT p)

so that its negation is

NOT (p IFF q) = NOT ((p OR NOT q) AND (q OR NOT p))

= NOT (p OR NOT q) OR NOT (q OR NOT p)

= (NOT p AND q) OR (NOT q AND p)

The statements in bold match, so p XOR q is indeed equivalent to NOT (p IFF q).

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