Question: BTVN
Bài 1: Hãy so sánh $2015^{10} + 2015^{9}$ và $2016^{10}$ .
Bài 2: Hãy so sánh $202^{303}$ và $303^{202}$ .
Bài 3: Hãy so sánh:
a) $8^{5}$ và $3.4^{7}$ b) $4^{30}$ và $3.24^{10}$ . c) $37^{75}$ và $71^{50}$ .
Bài 4: Hãy so sánh:
a) $5^{23}$ và $6.5^{22}$ b) $7.2^{13}$ và $2^{16}$ c) $15^{12}$ và $81^{3.125^{5}}$ .
1. **Problem:** Compare $2015^{10} + 2015^{9}$ and $2016^{10}$.
2. **Formula and rules:** To compare large powers, factor and use inequalities.
3. **Step 1:** Factor $2015^{9}$ from the left expression:
$$2015^{10} + 2015^{9} = 2015^{9}(2015 + 1) = 2015^{9} \times 2016$$
4. **Step 2:** Compare $2015^{9} \times 2016$ and $2016^{10}$.
Rewrite $2016^{10}$ as $2016^{9} \times 2016$.
5. **Step 3:** Compare $2015^{9}$ and $2016^{9}$.
Since $2015 < 2016$, $2015^{9} < 2016^{9}$.
6. **Step 4:** Multiply both sides by $2016$ (positive number, inequality preserved):
$$2015^{9} \times 2016 < 2016^{9} \times 2016 = 2016^{10}$$
7. **Conclusion:**
$$2015^{10} + 2015^{9} < 2016^{10}$$
---
1. **Problem:** Compare $202^{303}$ and $303^{202}$.
2. **Method:** Use logarithms to compare.
3. **Step 1:** Take natural logs:
$$\ln(202^{303}) = 303 \ln 202$$
$$\ln(303^{202}) = 202 \ln 303$$
4. **Step 2:** Compare $303 \ln 202$ and $202 \ln 303$.
5. **Step 3:** Approximate:
$\ln 202 \approx 5.3083$, $\ln 303 \approx 5.7137$
Calculate:
$$303 \times 5.3083 = 1607.61$$
$$202 \times 5.7137 = 1153.37$$
6. **Step 4:** Since $1607.61 > 1153.37$,
$$202^{303} > 303^{202}$$
---
1. **Problem:** Compare:
a) $8^{5}$ and $3.4^{7}$
2. **Step 1:** Calculate approximate values:
$8^{5} = 8 \times 8 \times 8 \times 8 \times 8 = 32768$
$3.4^{7} = (3.4)^{7}$
Calculate $3.4^{7}$:
$3.4^{2} = 11.56$
$3.4^{4} = (3.4^{2})^{2} = 11.56^{2} = 133.6336$
$3.4^{7} = 3.4^{4} \times 3.4^{3} = 133.6336 \times (3.4^{3})$
Calculate $3.4^{3} = 3.4 \times 3.4 \times 3.4 = 39.304$
So $3.4^{7} = 133.6336 \times 39.304 \approx 5250$
3. **Conclusion:**
$$8^{5} = 32768 > 5250 \approx 3.4^{7}$$
---
1. **Problem:** Compare:
b) $4^{30}$ and $3.24^{10}$
2. **Step 1:** Calculate approximate values:
$4^{30} = (2^{2})^{30} = 2^{60}$
$3.24^{10}$
Calculate $\ln$ values:
$\ln 4 = 1.3863$, so $\ln 4^{30} = 30 \times 1.3863 = 41.589$
$\ln 3.24 = 1.1750$, so $\ln 3.24^{10} = 10 \times 1.1750 = 11.75$
3. **Conclusion:**
$$4^{30} > 3.24^{10}$$
---
1. **Problem:** Compare:
c) $37^{75}$ and $71^{50}$
2. **Step 1:** Use logarithms:
$\ln 37 \approx 3.6109$, $\ln 71 \approx 4.2627$
Calculate:
$75 \times 3.6109 = 270.82$
$50 \times 4.2627 = 213.14$
3. **Conclusion:**
$$37^{75} > 71^{50}$$
---
1. **Problem:** Compare:
a) $5^{23}$ and $6.5^{22}$
2. **Step 1:** Use logarithms:
$\ln 5 = 1.6094$, $\ln 6.5 = 1.8718$
Calculate:
$23 \times 1.6094 = 37.016$
$22 \times 1.8718 = 41.18$
3. **Conclusion:**
$$5^{23} < 6.5^{22}$$
---
1. **Problem:** Compare:
b) $7.2^{13}$ and $2^{16}$
2. **Step 1:** Use logarithms:
$\ln 7.2 = 1.9741$, $\ln 2 = 0.6931$
Calculate:
$13 \times 1.9741 = 25.663$
$16 \times 0.6931 = 11.09$
3. **Conclusion:**
$$7.2^{13} > 2^{16}$$
---
1. **Problem:** Compare:
c) $15^{12}$ and $81^{3.125^{5}}$
2. **Step 1:** Calculate $3.125^{5}$:
$3.125^{2} = 9.765625$
$3.125^{4} = (3.125^{2})^{2} = 9.765625^{2} = 95.3674$
$3.125^{5} = 3.125^{4} \times 3.125 = 95.3674 \times 3.125 = 298.02$
3. **Step 2:** Calculate $81^{298.02}$
Since $81 = 3^{4}$,
$$81^{298.02} = (3^{4})^{298.02} = 3^{1192.08}$$
4. **Step 3:** Calculate $15^{12}$
$\ln 15 = 2.7081$, so $\ln 15^{12} = 12 \times 2.7081 = 32.497$
$\ln 3 = 1.0986$, so $\ln 3^{1192.08} = 1192.08 \times 1.0986 = 1309.9$
5. **Conclusion:**
$$15^{12} < 81^{3.125^{5}}$$